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The axiomatization problem for fragments
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!i~ IS], which we :~,~ad ~efer to as O~\F. we some axiomafi.zatkm,: resuI{s for intuitionistic theories, q"hat i s ~!~'% ,--*,. '"'. ce.rtah"~ theories 'I'~>T> with T~ a:l exte,/~sio'a 0{ To ir~ a larger !a~euave, we dete;mined an a:xl.omati[z,,ti0n T ~ ~f ihe addifiomti cm~se~ue,nees Tz i~ad over T o {n the ~r~:m¢~o ~'~ <: [am~uaee. Put differemtly~ We K~0ved some c(n~serv.aii~>r~, resuits: .~,m~<>~y'i'~ with certain structure is conservative over ~,.~,..jel~,~r,'-0 ~t'~ 'w'i,~a~tL.~L~v,.the ~:~ven' ~t~a'-'m.~co .~>.p......... "~1~2 ex:e~di~,?: a gweu "?[~ hv a good choice ~,aI *" AX{OI11S, tf~ t[~e o r e S e F l t - o a ~ 3 * ~ ; w e d o p:,e~ty m~ch the s:m~e tEim,,~,.........but' difTcre~d,¢,,, Tl~e re,aso~ f!or ttlis new paper is our v,,,,,,.(w;,,4 ~md,,r~*a*~d~"e ~ f ,~,~, o f " , " technk V es used in OAF. iL:ssei~tialh,, ore: method of pr0vin~ vhe co~servatio~a of' T over T[~ amsisted, in showing fiiat the eiemmi~:s ef a representath.% ,?'[a~,~ Of Kripk e models of % w e r e :compatible wkh t h e additloi~a[ Stri.~ctmx, re~:[[th:ed l)',;: T > T h e representative Kr:ipke m o d e l s used were, ,but f0r stJght
0':~......"o-v
those obt:aii~cd from aopivi~ig.llenkir£s complete~.ess proof to Kr]oke :models. As an. a.~'lerth0uglit . i w e .m e n t k m e d. that ~me coukt view Our m e t h o d ,:~.... ~,,,~ app,.~,~,:.~,,~,' .... ~':...... >' ~ 0 f Saturation properties. A f e w m o n t h s after O . A F w a s accepted: f~r m~biica:tio~a, o
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[1] appeared, 1The ,~':~ccess of Barwise's unabashed Use o:f saturati6e -properacs m the obtaining axi0m,a*izai!t<, resdlts iti classical I0aic tir0wpted: us: to reco~S~-der " " ~x~teo f Satl rat[0~, in 0m" ox~r~ worki As we me~tim~ed i~: O A F , Our proofs co~.~d :~I~ be refo~,mtl~ttq:,| i[~ ierms of sat~tratioa !?'ii>.)~v,t,.~a?": ":') ,_,.,dqFsa.,¢&'"".~:'°i;""' ~ '- ,,~.,,.,...,,,~'°;',"~'-'~"",~v~:o, t"7 moreover, tilt" 'aturatiot~ :prc~_pertieS m:eded are a bit more stai!dard t;Ea~a we su~ec~tedl Ti::[C,.~ ate, howtwer, s;it,.tratidfi properties 6f Kripke n-~odetS wire ~¢,:~ aaat0es for sin tile :CIassiCa.l m0deis'; : : [ I1i S e c t i o n
a~:~d sa:tm-ated inc.,dens. Fi~e exisience proof iS far from novel..but must be mctnded becai,Se o[ the: X'e~u!t~, ~ot Obtai:r~aMe :by appeal tO ."he Classical case~ Seetic,~t,s: 2 a~.d 3 a r e devoted :F,applications to:: the:, axi0matiz~tion pi:0biem; if~:tiiese.two ged{~oris{ 'we rei~tir~c~i)ti[~¢O;t}s to:a
1, w e
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disct, ~ ,~at'arafiO~t propetfies
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adVaidage:thai/~ve*:~,t~ ~i0w refe:~ tLe reade:6to gat,MSe's paper tc (:,,bta,i~:S, betie;,~:' ~}~::~sp~:ctiX'e'for the ::s.i,**!o f r~}uh:: obi:aiijed :and their so:uree~ '!1'~bs6 b f us[ ,~,.h~yac e
!94
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take,: wi~:h rhx' .... ..creater ex:,~ressi'~.'e ~x;,x~ or. ,,~:_ . tile . {~uii~ionisdc . . . -.u.,-w,,,e~:o...+, . .w.f:~./:. . . find ~he a d v m : t a g e offset b y the reaiizatio~ .h~,, axiomavi:c.:tio::s hi:tee ,-~4~,Sjs ( > i t h : u .~ slmtc,:er, a n a l o g o u s .:~ ~'" "'- p:oo!h) ~n. the cIassicai ......~" ........ .,.;.. ~.a .,~.~ e.~:v .'*~'-"" ~an_m~a~. rich m~ough :,~e? ~2 ~ to begin ~.~i:~:. So. s o m e of t~c mvst.e:v _ . is ~os:. At:caller d i s a d v a n t a g e of ~he p a r t i c u l a r use m a d e of s::turavc ~ m o d e : , is tt~e u o E n e ~ of the a x i o m a t i z a t i o n s , t ~ r ba,ancw, wc otter S e c t i o n 4 wit:: s o m e sin:pie, ~mt :)!e?.smg. results o n e x t e n s i o n s of the t h e o r y of a d~.,.,.aaL,.,~ eqmv:ttvl T h e s e tim?t! I..s:~t .s .fro n o t require, a n y s a t u r a t i o n at am" '~ b u t ~:rise f r o m considerat:o:~s o f Sectio:~s :2 a::d : . S e c t i o n 5 contai~:s s o m e e~.ene,'a~ r e m a r k s ::nd a few o p e a p r o b l e m s . first sppffcatio~ of the tech~:~iquc, we axiov,'mt'i;~e the co, ..~e~mc:~ck:~, - .... -" of :he ~xi.,;temx of a choice function:. Ore' proof is esse::,m~!y *" q 0~at 9,'e gave {~10,,k[?~ ~"e. 'We first consideer a transitise g r o u p of automorp::is:~ns a::d axio:::atize their conse-que::ces by a s c h e m e of exte::dibie h o m o g e n e h y . T h e co:~seque~::ces of a g r o u p ,~pcratio:: a.:e ~,x:t ,a,:~:,,.¢c ,,. a s c h e m e bes: d e s c r i b e d as bacK<:ne:--vort,: e x t e n : ! b ble u n i b r m h o m o a e e e i t ;., , (with eomposffio:: .. mev~lam~m~. " -~ . . . . ' T h e r e s u h s of these ,wo sectio::s are: p r o v e u i:: a g r e a t e r genera!iQ, it:at is s,.t~ticie~:t to see that t h e y are ide~a~{ca! to results . of classical :og:c-~-p:ox~dcd ' " .... o n e has a ~i/fle: mole, to the .
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W e hesit :tie to ca!: the p r e s e n t p a p e r a s e q u e l to O A F . As~:e " ::on: ~" " {he fact " 1!mt we n o w have a vasfly m o r e sopilislicated u n d e r s t a n d i u : of the situatioa, w e are <,.,,.~::c,hesitant of: f!:e simp!e ,.,:rou:ads . that we arc: at tile s a m e "',,ma,. .....wu~:..~' ' : ; -,~ :: sort of i~'~tr~itio:~isticaih:r.Oriet:ted scqt~c.~ ~o .i~.!, N o n e t h e l e s s . 'h wili bc co~:we::e::~ :~ a s s u m e ::~.c readc:" familiar with s o m e of ti~e mate.ria: o n the Cc~mpleteu~ss T h e o r e ' m r e v i e w e d h: Sectior~ I of o u r e a r l i e r effort as well as with o u r sta,udarc~ n o t a t i o n ~.c" 3~:~.,:.~u,m~." 2: Kr}.Ig e"~9 models, tt is atso convet~ieat to a s s u m e tl~e r e a d e r to b e fami!iar e:aough wffh K r i p k e .m~dels so flint, w h e n we assert, s o m e t h i n g u[×~ut the s t r u c t u r e cf the t?~ica~ m o d e l of s u c h a n d such a thco,'v~ we n e e d ~>~t s u p ! q / the trivial xerificatiom Familiarity with I3mwisc's p a p e r is n o t necessary, a u d m a y n e t bc heIpfut for specifics, bu~ is r e c o ' m m e a d c d fo~ tb.: b e t t e r p c r s p c c i i v e offered. Ar~. or~1,:'!©~:ica{ dit:ferc~lce b e t w e e : : the presem: p a p e r a n d tI!0se cited s h o u l d b e ment~.o:'~c,d: ~an,n.. v,..,.,,: ::::'eviousty everythi:~g was cou::table, we n o w find it conve~:ient with ~:~,,~
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o~ Kripkc m~:!~ts
The.re are two w a y s of com~tructim~ s a t u r a t e d Fh-}:~ke ~:o: ~e '.s ........the c h e a p w a y a n d tI~e s t a n d a r d way. T h e chea!~ consucucti0~ consisu~ c4' redt:ci:~g t h e p r o b l e m to
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iO ~,4,@.,. ~::Odei:S, ~-~01:t~! c:,£W so:!~td 5:~:a:,:a:}:, t h e ~:',:z:)co~.s-:!:r~c:t:o:is a:°o sot "T'[~8 C i i i ' i i ) :''4~"~'' ,r*'~';<.o~ " . . . ' -p:<,<;::c~,~.'j='= A= ......... ==".....
tho
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T is ~<,[<.~,~ Q,"""~> ,i-=,=~,class o f 4"<~d,:~L< <';~, ;:~: ,~.£,-.,,.~u,e<="~ ..... " ~t,~,,t-~.,.,.~.:,~,~ c~s~,g~cl.:! tb.eo."y '}7'. @'Or ':. . b.{~ . . . . . . . .to. . . .~ c ~~c'.q:{~S, se<~ c44,." {7. p . . 3 .~.;. ". .,,-]~,-rq.:[>~,. :~'8{;#aC£;~0>." " O~ ~I;~11/ ::#.-.Sv:'[S'01~' K . i : ] k e IY:Qde~ t'hc:ory i o
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have, however, bee: u::ahie to -o,::,-:tXe cheap co::st:cucdo:} to obta::-: aX +he St~'['vl:2a::10~ o:'o~.81't}e5 We ~,.,~o '" • ~ ::sef::.1. }-:e::ce. w e h:sve ~: c o : i s : t i e r t~:e m o r e cos::~y s:a::dard co:str:~ct:O::o ] \ w o r d o r t w o be1~ore ~'e:,8.::tic c o n c e p t , S a t e : r a t e d theo@~s a r e .~"~' ,"-+;'" as a syntac'/@ -~ather ":ha.... t h ? a c(mst:ruc~ed as s a t u r a t e d m o d e l s a r e ciass:calh,.. ~.%=t,,,-:~,*;~=, ....... K r : p k e n:.,de:s',-'" .... are t h e : : v i e w e d as p a r t i a l y o r d e r e . l se~s o f s::-tL-:ra~ed :::eo::6s. .... A ~1,:.<-{e~ ":'~'.- care :ii " d,-dr ............
cc::str~:ctio:,,:will yield.,addiHo:::.,isaturatior, pt'operties havi:~g nO {obvio::s) ar:a:o,~::
in t h e c:assk:a: case. Let [. b.s :: hmg,aS!:yco A~ :._,-.the,n 0, £s a psir (I: A) o£ sets of se::te::ces of :.... .
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is de~i,eable in t h e i n m i t i o n i s t i c p r e d i c a t e c a : c t l u s f o r f i n i t e [ " o ~ F', Ac, g ; A . A. cot;sis~e'.:t t h e o r y (.[~, ~t) is com~.:b't,~ if, f o r ewsry sente::ce. @ of L, @ ~ [" o r <# : ,~ T w o t h e o r i e s , (]':,-'A :} arid (.i"~', k2}. a r e CO?rigiS.teg1~ "Wi~]';~g,..,,~,.,,>.,;~@g::,~g~.1~ (.~'! L; . ~s ¢': ~ - ~ / ~ 2", l ~':, A" L} d ~) is c o n s i s t e n t . A t h e o r y ( / - , t k -) s,~ong,!y #x~end:: :.: % .,.:-~ ]1 w~.~ [.,ave boil:: F" :~ t " a n d . ' ' : ~ % tf w e o n l y h a v e t h e firs: : n c I u s : o m ~ ~ .~<~, .wo say ~'}'~a~ .... exre,~as ( [ ' L :Y:'}.. LQ{ C i:')e a s e t o f ( o i l s t a n t s . B y 1.40.), wi: m e a i tl'ie Iar:g::age ' *-" ": " ~'7 t o t.. By an L - n - g y p < w e m e a n a p a i r
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v . *.a:':::o~e:s o f s e : s o f f o r m u l a e o f L ~ m s s e s s i n e o n l y .......... m,- f-<:,e , " "'~ i~ted arzd such that ,.1 .... : . . . . . q.,). "~(C:( ....... q,)., is a cor~sister~t L({C,I... <'i}5 eor'-,": 2-" .... c;, a r e d i s t i a c t n e w COI]St:{t[ltS. S" iS a. type.C;v*ir au L-:thcorY vI <~', k ' ) if .r(c(," . . o , c,,,.) ,!s c o n s i s t e n t w i t h (./:', 4'), i:~ *-,t~.:, ~ *~ . , . , C,,v~ >' t h e C~'S a g a i n be:::~, d : s d n c t s e w i
[ 96
(Z S;:<'~ ", #~:s;,f
coastm>ts, rix~ . . . . . . xo~) is ~ea~izcd h~ (.F'~ A') by cor~ste. ~*s Ci . . . . . c: ~ L ~i:(17, .~.!,) strongly exte~ds r L Q . . : . , G)~ ...... With these basic defhfitions, we ca,a begL,: te discuss sat~t~ti.e~ properde.- of theories in earnest. Tt~e weakest notion of ¢a{Ui:atkm is that arfsin£ in the:proof of the Completeness T h e o r e m asserting the existence of '~,,imessf~:~g co.:~lsIants: : Defirdtion, L~t (F, zl)~'~) a complete L..(C)-theory. (K A):~s C~ss*~.~>g~-J if eaLcb
~.k2 a mode!.otheorist, C~sau~ratioa is a bit weak to be~tr s~ch aa cxa t~d r~ame. However., too&fie a mi~or exerci,"e to verifv {hat :it coi~~cides with ihe usual defiMtion of C-samrmion for i~Kuitior istic fl~eories (as iu, e.g., OAF), the term is fairly well estaNished ia the !i~erature ar.d we shall ,-e.,ai~ it Ti::is over[: U) c4 intuitionist:[c a~d mode~thee'°~!ic tcru'~{nd[o~\, is r~o~ ~i~,~.'.~x-5to cause m.{:~ch~:
lqrst Flmdamenial Exislenee Theorem. L e t ~,:* = 2% L a h?,ng~.~r~gc,-~j:cardb~,di~y +¢.
L..-theoe),. 77~e~ rhe,,e ~-~s a .hd~y-.C:-sg~.;tm~ged L((?)-,gh,'.°o,:y tl'~-k ~) d:a~ st~v~aggy extends i f ; k). t r o t L We couk~ obtah~ this result cheaply; D',t ff we are mv:m~: the ~tan&~M construc~o~,. we shouid be staadard *t~,.,u~ncu~. ,~,~ { -~~* Hence w:'~~ ra'm~c the class{ca{ p:roof. i o t L-"~ . . and { , ~,,g~},,<~, enumerate. C -:;~:Id tLe set of dis}u[~cto[~s of :~e.~e~,c:~; o ~, L ( C h respectively, a~d Iet '{5,}e~::~ enumerate the set of alI LJ._.>I*ypfewer tl~au ~~ Coustants oi! C. We shai~ ' ~' " ~"* .::1",, by
k:SS thatl -~+~ Assume (.(~,, ~,,) and D,. have bee~ defi.ed f0~........ ~,'~ ,.<:::~-,~ Let
OL0{ e~ ,_~,~z1:27 G{:)~/f ....
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by
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the crucial t)~;e-:~:eaiiiz~tior~ c}ause -of t h e ~,-,....a.,-o:~5d,.us~ is c o m p l e t e a e s s ..... ,:;,.,~,.~ ,. with its s u b c i a u s e of cous~ster~cy. "M o d u [ o co~?msteac% c o m p l e t e n e s s holds bY the defiMtio~ Of ,&'. Consis~:e,~cv h.h<.ws ':,.~'- - ..... f a ...... ,,,.. the eo~>:<'f.~,'m~ Of
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T h e ~:hst F t m d a m e u t a l Existe, nce Theorem_ ~,.~¢vo~....uS a !a:~'ge. >.o..<* e" cz~ f ~...! y s a t u r a t e d theories, i n classical rnodei .theory, this is [he s a m e as the exisi-er~ce 9f s a t u r a t e d m o d e l s : Iu K r i p k e m o d e l theory., a Iittle m o r e w o r k is required"" - - buS" ~;ot much: A K r i p k e m o d e l can t ~ v i e w e d as a c.ollectiot~ of s a t u r a t e d ~heo!ries; ~:, " C o i e c t k m of f u l v~ :_~i't a ~ : u"~ th(:::O1Jes, s a t u r a t e d K d p k e m o d e l wi![ b e given a s ~:,,. " {h~"~........ Sc:me ~lotatiom L e t K = (K, .a~ , D.. T'). he a K r i p k e s t r u c t a r e for ~~t,~ Ia...~+>~,.~_-~,,..-,,~a L A s s u m e : we h a v e a s s i g n e d set s o f "c0nstgmts C~ n a m i n g the e I e m e n t s of D ~ acM t h a t the a s s i g n m e n t satisfies file o b v i o u s c o m p a t i b i l i t y e0~Mition: {f G ~ ~ and <,, m~mes x<~ I;k:<. r h e a c..{5. (~,. a n d G still ~'mmcs x at ,~.S E a c h a: e: ,.;< detennh-~es a G , - s a m ~ i t e d L((I~ Ldm<.,l"y (!~. &,): K,~<.
. . ~ "" . -. " . ~ ." / " D e f i n i t i o n . A K d p k e m o d e l K is ":..........
,.... sm.ura~ed,' '~ is mlW.,,Q, "
. ,~,~0~.~, AA<, O u e ¢7m a r g u e l!mt a s a i u r a t e d K d p k e mode~iS!~oUkI *oe t e r m e d < ~,, S a t u r a t e d K r i p k e m o d e l to p a m t l e l the futl s a t u r a t i o a of the theoxies at e a c h o f ~Js nc, deSl Such t e n n i a o l o g y i s :Un~'arabiv inflafio~mrv: N o t 0 n l y Wo~.:iid :~hat = ai,m sue:~es{ C a i i n ¢ all Krit~ke = m o d e l s S~it{;fated (:i:0 p m : a l e l :tlle"sat,aration Of the
d[
7
Detlni~o~. T h e IT~crea:vi~ ~r.?;z:~+di,+v_z~(;2~,~&rb:~ oi:~ a .b~r~p:~e m o d e l K is the following c o n d i t i o n : ICC. If a < ..... 8. t h e n t.1~, camd~nalkv of D a ' iS . '""~ ~v.:s. -*ha~:vthae of D~J. E s s e n t i N l y t h e r e m a i n d e r o:f the sta,adard cousmictim~ col:isists in e:'..~p}eying .',t;e F h s t Fimdame~:ttal Fxiste~~ce Theo~'em . . . . . . i~. ii:,:'.. <~,". ~.,~,v'," t c~i!the r',~,~,,~'~',' , . ,~,,1,,<*e,,-' ,~ ,,. ;, ' Filc,.~r~2In w h i l e also wot'~k i~e.,_{!~e I...c,,.{ ...... iv~t;~ 1he cor~strt~ci:io,a, l~hc 'u~;e c:?~'Ib¢ t:q~st: Fm!d t~Ic~>. tat Existence T h e o r e m w[l! g u a r a u t c c salltr:ttioli; tl:l~: prOOf 0t7 the C o l t l p l e t c n e s s T h e o r e m will add a n o i h e r saturati0p~ p r o p e r t y {not nee',~ed in the p r e s e q t i_:a)cr:~, a n d allow ~or t h e satisfaction of fl~e ICC:, ~,~~ ,':he t :C,~ in the ........s-,~ .... o~: sat~e'ation,. wilt . . greatly . . . stren~the~l . the p o ~ e r o~~ saturation, > b r the s:i~e ..... of breGl:y, we s~ate t!~e folk~wi'a#~, .~ T h e o r e m r a t h e r weakiy:
S ~ o n d F u n d a m e ~ ! E.5stence Theorem. tl_I-LH,. L ~ ~F:.A) ~:e a cc,'~s~:<-e~..: L.,U~eory. 77wn (F, A) has ~ san~rared ,~;,:odd sa:~is~ing the ICC. P r o o f . F o r c o m p u t a t i o t m ! d~nmt~:a~...x.~, ......~:" * ~, ~¢ let L b e c o u n t a N e "rod let Lb~;~ -' C~ ~7..o • be sets of cos~s~,~nts such that C,~ has ca:r{~m~.hW -" ~ c~ ~" tic,, '.... A,,~ ... be a fixed ...... m~x,. "° " ~ N,.~.~ ...... ,7 e--I ,~ c.o--satte:ateo s t r o n g extensio,~ of (I2 J)° W e defiae K = (K; ~ D, ~b.'} as fol~<';ws: K is the set of fie.ire seq~ei~.ces,
such that (i) vl'o, ao., ~s the t h e o r y chose** ~bove t.+.nj ( ~ , ~ ) ~.S a fttlly-{~-saturated L(~)--Iheo:ry. T h e s e q u e n c e s in K are given t h e u:mal pm-tiai o r d e r h l g of iinlie s e q u ~ n e e s an First FUtldametltal Exis+,eitce T h e o r e m . the us~lal p r o o f of th,~ Completene:sc: "i' ~
ia (*),
wbe~me (F<,:,& ' ) = (F,,, .&) is fulfv-C,,f =~:i ),:~'>s,amra~:ed; w h e n c e K is a s~mm:~ted m o d e l of ( E A)c2; (Ir)~; Ao).: : =' T h e I C C h o l d s trivially° q + e . d . ?
S'c'~' .~i~g~!~:~,~
? , h ~ "~.,W~ ~-~.~:~'~
~,x4
to~:" ~ ............: ........ ~i:~"
~ K '~-~ b >
i 99 °
. .
\Ve ~o[e ,,.L~ [,.oL,£:.., A~ sat~.~'at~or~ pro!::.*~;~aes of Z~[:
.~),L'g~
:
(
(!~,
"
k:. )
'
'
:
....
{;i~
"J'a), "" %
"-r-~
......
, ........
b y s o m e ccms~:.>::t i~ ©,:>.
K a~id the. '{CC:: 1-:<~r S < a' t~e c:.m1{-,:ai;~.:~ (;( ~="-~ . . . . . .
a~Ao :we
" :~.,.,.Z,:.~:,.t:....... .c>Z d~e %'I.... }q~.al!.,.:~ Pro~0e:rty [i~i ~s a~: :........ a ': ....... co~tseqv~ef~ce of fi~e co~~s¢:r~:cdon Used :i~ t:~...~ comp{e.teness p:'oo~:: K c:~ is as ~,~ ('~;) a b o v e , we ca~: t~-ke
w h e z e ([~,:~>-;~~t~>~:) i s a p y {uIly-g~,.~.~saturated stror:g exte~sio~l o/: :(~" ~i'~I~'r(c}.,
cec,.,+,-'(.,..
P• r o p{..... e r l y.. t i I. is .a geaerali>-~tion oe the ~r~*, .....~Y • u~.e~, ,~.. 4 i~ o:tw p r o o f % . . . .0. . . a. . F (.r[ the . . . }.'-~ e'~-~ 'va,a t ) a t e n - S , a t m a ~ Tb:eorem. It wiI1 not b e used i{~ t!*e prese,~t ~>a~er a~d iaeed ~,"~ be d i s c u s s e d f~rt}':er Prope~:ty I is basic a n d n e e d s ~o f,,:~rd~er e.,@a~atio~:~ . . . . , ~hat a! "cadv' ot~ered }"rapo:t'ty i t is fh~ c~"acii:d saturatio~ prope~'ty !\9r w h i c h we gave the sta:<'~dard consu:uction. Whik~ F r o p e r t y I a n d a c a r d i n a l i t y - r e s t r k : t e d P r o p e r t y iZI .....'~.' a m o m a t i q M / y in the s a t u r a t e d m o d e l s given by the c h e a p col?stt't?ctio~l, P r o p e r t y i~[ d o e s n o L W e wilt use i t in ore" couslrucfi('ms ~:o a!low ~,~ to cc:mpb~te tt,~ c o n s t r u c t i o n at a ,:~iven levei of a m o d e l b e f o r e !?avi~-~eto c o n s i d e r ~ bi~4~e~"!e'veL
2. Const~luen¢~ of the e . ~ t e n c e o f a choice fttllc|ion I a classicai logic, the exis~e~,ce o[ a : k o i c e fm~cfiO~{...f~ VxR*t,.:~
: :
is col~Se..r'~*ative o v e r tile metre e:x:iste~ee assei:'tio~:l~ : Vx ~vR;vvl . . . . . . . .
•
:
With: intuitiol~istic loaic, the s i t u a t k m is m o r e c o m p i i c a t e d : Esse~.:~:da!i){, ~ae .:ea~,o~ is t h e m m d o g i c a l n a t u r e cf i n t u i t k m i s i i c equality, :I~t,aitioaisi:ically,' e q ~ a l i W i s
oftt-r: a d e f i n e d eqt.:h,a:e::c~ f,qa:.ioa:. F,.g:.. ::~': 6}¢ ,"~m:s. ,.icwed :}s co:}s::m:.hv~"
("~'~'1,,' S¢.q:ie>~C:2S~
{t :8 ,
~,. ~-*-..., ': ....... "- ....... :*" v
Of .... . v : "~ ? ' ~":"
~:itloliL NO[ ev¢IW, ftIItCt{O[: O:A eo~'IS[l:[i,'.:t{v<: C a u c h y s e q u e r : c e s p : ~ s e r v e s "h,:~:. .<,:b-~:,~:"v<' ....... rdatioru
There
are
ex:e:~sio::a: a a d
:~o~>e:,ae::sio:lal f::nct~o::s~ If t e e
c::o,~c:e
fu:~c~ioe, f a b o v e ~s ~:o: r e q u i r e d t o b e ex~e~:siOnaL i.e. if o n e d o e s : l o t a } s o a s s : m a e t h e a x i o m o f exte~siona;~itv,
v x y ( x = r - - ' . i X : B')-. tile:: t h e co::se::vatio:~ r e s u : t st:lit ~:olds." Co:.:se:va:io,,: -~~-.L,_,~--,~ : ' , ,,.~... v ~xh~,..:~ , , w e assm:-m ..... ~ ...... is it?ice, eo~:, dec:{<{ bo th¢: ~'ax}om o:e ,~x<¢..k,~,a.t~-.-prov:ded tg:e:e:ma{~ty :¢. "" ~a,: "~%-
b:e: Vxv(x=y'~,'-x=y). y e..',~,. . . . . . . a. "~,~ ~,'
"
o
::ecessa:°vo. a d d i t i c m a i t~_xioms a l e ~.-:~-~..,~.:1""'b y a schcm(~ e x ~ r c s s h ' , a {he cxi:~'~c~ce .of
exten°,.m:e th:ite ex~c::sioaa: choice: fu::ctk>r:s:
:,i ~
. .
.
.
.
.
aathor (OAF~. [ W h i l e
.
.
,--:~ .<.:,>,<.~.:~ aItd a l m o s t
,.
M i : : c h a s am"~o:mced, h:: r e v i e w s ,, "~ [.r,'~ t h a t h e a n d e~e
_ .... ~ s t r e s s th,~t w h i l e w e k n e w a u t h o r h a v e :~,mv:ded "d .c:,s ,a .:-..-,-14?~¢.:~n w e shot. . <'-, ~-{:~:~~ec©.~ks{iv o ~ :},:avL-~.a t h e i,l~:i[Fix sho'~,r~L w e "~ve1~©sio\v t o ~e:'di~=e t i l e s i ~ a i f i c a n c e
of t h e ¢x:e,.:dA:l~ts, .......... : ',~-. p r e f i x - - i . e , t h a t
o::e
an:is{ a l t e - n : a t e
qua,ntifiers
and
m.;,{
m e r e i v a s e V > : : - . . x , , 3.:,':" " "y,..i T h e n e e e s s : : y o f s o m e a x i o m a t i z a t i o : : .........Lc. t h e {IO.* l - - ~,'" OI
~-~ =S<~ . . .X'. .d 'd"<"~
t
of the existence of a choice flmction ove: the mere exis:e::ce . _
assert{o:: ..... w a s {irst d e m o ~ s t ~ a t e d
ia [4]. A s i m p l e r e x a m p l e
w a s o f f e r e d h: [5:].
~: :.,.:,c - * - u a e d as h~s .R. a {'or:nmla it; t w o too:indic predicates and a propositio:xal
col:slat:t: ( }:,:s,::vak~ ~:scd ~:~,h.,":",' ,,: m-~:',a~,~:'the ];,)::i::m~:c . . . of . eqt>n:ity . . . a:,w.mc:itcd by :~-::-'~.,::.~ co:~stanm. ::\::yo:lei'am}[ia:";vit:h:he i::'t:it:e.::istic theory of cq::al{ty k::ows that it ~s . . . . . . :..:~.........o e.u,?ugh wi,q:out a h t h i s a d d i t i o ~ m ! s t r u c t u : e , h : d e e d , w e o f f e r t l l e Counterexample.
L e t E Q d e n o t e t h e p u r e f l : t u i t i o n i s t i c t h e o r y o f equ:alily, T h e : :
,~X.,r + V x ( 7 x : . t : v ) + V x v ( x
= v ....... Lv - ': <'
CO :IS,~:f V::{W/l: over El.? # V ~ ? b ( " " x = y < z
t= .....("
Vx = "~ y ( ' T x = v)t~V..v: 3 v : V x , ~ v .
"
m : ~ =::
{:,:,.-TX>
::
Y2
:
"i,~],
P-~£ ~
.K'l;k) k~t
- = ~. . . .
"
(; ~tg~2%~ ~,G.~ ?~'Z~:)'/S,~ ,t), ~' ~ k ~ £ ¸ 8.~ ~
S [ ~~c(:'-'
:5~).~
. . . . . . . . . . . . . . . . . . .
:iS~=~a]iV ~Wl ~;(]uiv::lliellce r e i s i ~ o n , v~,.,,, D..4~.= ......... ~_,=,.,"~.=~,_=q~d.,=HU:" " : ' I'X_,.,:' K.~ k A e : K~o~Rel !~'~: m u ~ ' £ o f eqkl~vaic~ice rsiaIiot~s ~>~,.o',a t h e c2i}mlHas" " i)~ ~.]it
tl~,%.
=r~
I)&TWiP/
'] t:,-: : m i , , [<-)qk:~.ire~:,erii" " " o ~ II~o~7~" i,s
~}6rsKY~+II;,:
:'(Vi*:~L *; ,~ i.~ d. d. k.i o. ~ l a l t,,,:,~,:.t.,.,,, ~.[.o aim s,'~'wl.*i'--o,'-:q':>:~ e,q~,~iV~(!il#~]Cc~ "~,.,,=~,... ~q':;+:'}"~ . . . . . .<:,}s ~ ' " 1: "b~: = i}iiitsi co{,:erttense.
i.c;. slbstitl_~t~.vilbi . l'e[atlot.s. . . " ' :I' " °l%nt}iio "
8~.: eq',~.raii~y, !..... 'be = i inc)d~i~.,s I t t h e l °;
e. i_~ /)o :. :~ FormaH
7. £
iS c~e:~..-t.d ,."v"
C,
-~. t q,
1[ iS dC;.ii'iV 3 IUO(i8] ° O[ ! , O Vx 9y(-l
3.i?.(J, ::dilCe " ct,t~, -' ;-' -]'4;" = I .' -SD'. = / : , R is a l s o Ik ".-~a,!',el ..... :,:
x == y } , M o r e o v e r ,
:
a%I/'Vx~ Iv~.. Vx:~ Iv:[--'~.~ t ,:= ::,,,", -r,-.,. =. .V>A(X, . . . . .= . X. ; -..*- ,: ,, -- ) ~ : > FIt',
aSsm'ne~ tl~c o p p o s i t e a.t]:d l e i :~.:~= x+. b. i . ~, . I , ,'esPecti,:sly. '~-~"=~,.=:..,=y :
d. c. r e s p e c t i v e l y ; ~:¥;i~
y~ -rm:~si: b~;
but
=: b - - > d "= c,
a
si~lce
=x =,'..-= ~-~, ~ ; l : d
:-'=it,
We shall state our result in some 'Le~ T b e a n i n t u i t i o t i s t -i c with:cq~:atiw
atd
Ieneratty.
. g~a<',e~ L_ tIieorv.. t~.~ "'.r t h e i s u a l s e ~ s e } f O : , : m u i a t e d x~s a a ' : ........
l e t #,'_vl; • *a%.~ ~ b e a f o : r i m t a o f I.. wi[!~ d',_e ~=;,,~. -,,,,,.==.=u ,,,:is .::~. . . . . .
v,~, ~ m~d l e t l ' b e a n ~.]ai~,,: [uriC/iOn] s y m } J o l n o t i~;~ Lo DeVOtee ' P ' 1:o b e t h e i h e o r y :k~ i.. t;i {or} o b t a i n e d
by addin~
t o 't" t h e a X M m S ,
V.~= " ,..l.b,,." ~ ".-;'2 \1~'~ X Vx~, ' :
:'<:,:/4X~ ....
-'~:"~'~-YI' " "Xt ':'=:'X2
x.,. t'.~ .... ~,,:.
tl:~XJ
''121
.
: (t.. t: i
=
:
i
i
V:v F "* v', 9v," ' .V:v }- • -x.,'i, 3 ) ' . , [ ~ ~h.,
=:,
~
.
.
.
.
.
.
i.e. (P" h a com>,r--~i .... ~:vtcv~'s/o.,~¢;;~T+(-'.~ ? T J' exte~Ms T-.}: i ' , ,,: ~ , iS :t l °t Y.laL P r o o f . "I " ~~;0.t ," F o r a , ,a,lc.na: c o n v e M e t t c e , w e s h a l :~ssum{> ~>.a{ p R. L e t ~ b o a sense[Ice' of I., W e sh~;tl s!~ow h o w to co.qvc~1: :~ st~f~ab.ie s a t u r a t e d tree m o d e l of (T. {@) w i t h the ICC' i m o a m o d e t o f (.],,i ~.,.~ --~" 'IY~ Q~is_ e~d. Iet K = ( k \ ~ . IX I" ) be fbe ..... a,~.v~,c~'t o [ iT. {~}} co~istructcd h~ I~e p r o o f of t h e S e c o n d F u n d a m e p t a l g x i s t e , ; c e T h e o r e m . A:s ,aswd. m,, ,.tern tixe o r i < m ot (K, :~-.'). M o r e g e n e r a l y ,
.
¢>
=
T o ~.u~, ~ ........... .# ::md~i~.uitv. ..... , the i,~te, . . . . -f,. r. .e.t.a. t i o p of e q u a i l t y we slm~l use "'%, " to ,-.. l c no ,... at <~,. t hes ~ea.. es - : , " wifi'l ott y two roles: to d e n o t e i d e n t i t y a n d to be the f o r m a l s y m b o l in ]I T h c { e i:~ m u t e thau or~e w a y Io h l e l p r e t f:uncdo~s. ' I b cow,sider a t:mary (say) ~w?ct~on to be a b i n a r y r e l a t i o n S satisfying t h e e:xt,~a a x i o m V_vN! vS~.x:y:is p r o b l e m a t i c : It explains f u n c t i o n s i~ t e r m s of w h a t w e Mready ki,~ow a u d is as e x t e n s i o n a l as S. H o w e v e r , if we are ~oi))g 1o co~.~s~ruct au i~terpret;:~tk.m, it wilt b e ~..,.,,:e,-..~ t<~L)terpre{_ a f l m e t i o n c(msta~t by somelhi~g., l i k e a f u n c t i o n : T h e i n t e r p r e ~.:~:i:)',~ "" ~ is an i:,u:.texed family >e ~ , ae
~
-
,,:~: : Ore ....... .[9,:~'. s~tisfviag . . . . ~ ' c:: ,]..... T o g u a r a ; l t e c t h e a x i o m of e x t e n s i o n a ' , { h , (~ a n d ai! X; v t~ ,L~x.
:
i
}~ ~s ~c.cvessarv m*d suftCient that. for a]:]
{
form:
,
<~.
.
~
N*~ : ~ , . : - ~
~r@ 0 i "
yo
consta~t reatizing-,'~,.t,)'~'" A s ~,,d~,ctio,, hvpotheM~ we t~
where m is mi@mff~ such ~hat % ~ C",,. Sinc~, % has an e:m.;:.~L~'second com:oo~?~mt, it :[otiows that % is prcserved ii~ goh~g f ~ m a~, to c¢:: d~ reNizes % in (.Fk. A.~,.).
I ~v~:t V ' V ~ B v V ~ , ' ¢ ( * , , I°2(V)
{X%Y: Vx ~y,y=~y ~
}, f]~(%)~,
Cr'=O c > 0,
Let (r > 0 'I n.~. mc ~Otonicity of the .F:.,%, O < 0:=~ r;(d~3 ~ I%, (.%,,.
Lj,,~..~r~,£,:t,.~
readily vietds the consistenc~' of w:M~ . . . . " ' f o r a~y o , m s t a n t c ~ L((..,,). Hence. Z , ( y ) = (]~,.(y)~ ,(% is a t y p e o y e z v,. ~..,.,.~,~,,.~- ,-~:~' cas~" w h y ' r e ,~ ~ 0 i s
similar, but easier.
Si~ce ~:,(v) mentions Only constams c, fo: 0 s~:o" and d,; :fo~. .f,-. . o,~" ~f r~'~e~i:iH~r~s~t: m o s t N, c o n s t a n t s a n d the; ful~.oCT,.Lsaturat~o~ o f
~ "~'~ ~' "~:~" rea~tzaL~i~ ....... "~ .......:a[,, .....
'r,.{,'~?) by a cotlstalH d . ¢ ( , . , v L e t d~ bc~, su~2ll <* ell
Hence, for eve~:y q. ~: tA~,~ w e obtain =a d~ ~ 1A%, which ::eaI!zes %~,p cn.e:~: ; : " w:e i:~j:£!~ K ~' '-~ i n d e e d 0~er (!"~, A~, ") for all ;~ ;:~ ,:¢;.[ I_Mn,~ th~s ass~g.r..~me~L i n t o a model of (T f', {q{})b~," definin~= for each c% (I"
L (c¢~- =-'=d , - '
'
[
:
:
(~,
By const:ructiom 'each
axioms :of:'V: : C h o i c e : W e m u s t show
is a n exte~.s:io~ of ~:
wh¢~Ce ",~e E~a:~¢ ii~de:ed
, ....
:
~o ik V XRx~r.
c~ !~"R ( < ,
,Lc,(~
}.e.
,re !P Rc=d~.
B u t t h e f o r m u l a RGv. occ~ws in ,~,.~,,,,i}i~.e '~1 ae~d d, ~eaILre:s "~L "'Fi'm~/~ i"~c.. ~ . ix'.. ct I~..R," ,~ gxtel~sio~lality:
L e t ,,," b e a r l q t x , v y
Q , %, "*
~, . . . .a ".
e~ ;,~
"7 - L ( L -
}
coP, t a ] n s *~-" u~;, s e u ; e n c e .
RGd~ A
Rc,,.d...
A ( G := q , - ~ " ~ ..... 4 ;
w!ae~we.
c¢,~i> % =: <,.---~'-lb., = ,fi4~:~'Rlk,,
~"~" ~',' "~' . . . . .
C.~, . . \ r .e . a l.< o. h a v e
f * ,, L¢, < "......... ~; ~ ' , , ,e,~.w
q,e.
t:]~
.a.~ :.m app~icatio~ oi: T :, - , ,~ co~,Mder the ]ine:~r ord<>r ,.,~ ~ the ex:istm-~ce of :m i m e r v a l sefector~
1..~, ,~~. of
Defi~tlon. A b i n a r y l:uactio~ f (m a n o r d e r e d set is a n :in~er{::a~selector iff f saa,~nes the axioms,.
',~ .~..a. :~g a= f . Y V =-; V
~: > }..... X :> j k y :> v,
in~erva{ scJc-ctor is a e.euerall'¢~ f o r O<:,A < i~
mk!ix:d~:
Opc~uto~
More
i~: fiv}~:tkm.
is a~1 i m e r v a l selector in tim :eals~ F:om the a x i o m a t i c p o i n t ef view, a n u :~¢~,~;~ ,
for
the formula
~',:
x:i ,-',x:~~, x?,L :
i
,
. . . .
,~. . . . . . . .
i
,
'
the. s , , ] t ~ I ~ ~ ~.~ ~ £ ~
o~i-~ <;.
O~i !i;~¢ar o r d e r a~re~axlomatilzed
Q
r,: . . . . -~. ~ :: ~ ~0 ::,: ~ 2 ~ >
kw £he .~.~.~.,~o~4"~-, ...... ,~.~..,~,~ ~.~..°~.
:
~
~~
~.;:~:~ ~ •
]
A :midpoi*~t o p e r ~!or, *~: sadsfiet-. (at ~'........." ~.1, ,-..,.'..';~'. ,-.~.
" . .
,i:vy =,:/yx ~I'#1,..
&~&
$~(¢~-,, . & i - . - -
~:.....
] ~.d \='-~ .,&"2 ~'~ } ~W""2:' 3 , ] " - .f"~*]'"
mat:~",x o f t1:e a x i o m s c h e m e f o r a:~.~ i n t e r v ~i ae,,ector~ " tl~.e coRi~a.*..~ct~0ns, O~ ( x
.....x 7 A x'i; =: x --,-~.... )I
v.~ ~,
.
Li
resp,:?.ctiveD. L e t ~ s p v ~ s e k~r ~,~ m 0 m e ~ t
~o c o n s i d e r c l a s s i c a l t o - i c T h e p;:0e>f o ; Ti,eore~:~
adap[ts quite easflv to exoal~d classical models 0.5 ciassk'al . . . . . . " m o d B t s o f classical T"i The. f a c t that this p r o o f is not n e c e s s a r y - - that tLe wh~}-,:: S c h e m e ( * )," c a n b e r e p l a c e d t~y v X
pectdiari o,
ByRxy
-2. is a n o t h e r ina~ter. T B a t is,. d~2e:to a
othde ,Pea{Is~ .... : . ~v,~ ' ~ d o :,~Ot >ieed o?
o:f c l a s s i c a l equatitv.~, <,stablÂ,,u~-d '~ " "--~..... bY s o m e
t h e cOnq~licatcd: a x i o m s , tTFor a Sin:dim: e x a m p l e , s e e Ba1:a ' ~" 1se s cr~:er.,:","'~'c:&~ ~. ,< ~ [!:D to
Gre£orv's
simpler
axiomatizatiom
."
~'" : "
: ":; ~:
~-° ':-v......... ~;,.,~~
i t v. a l v a t h e o r i e s . ] ~ I f w e r e p l a c e : e q u a l i t y~. bv~ a b i n a r Y . . . r e I a t k m E " . an{,. .'exte~s~0~"~a' . . ..... .............. . hom0rphism
condition,
Vxy(!£xy-:.~ .E?: :ty,~', w c e c t c i a s s i c a i i v ~hc & ' x:~0B . . . .,~atI,.,a : ~ t~oD:, ' :
[
. . . . .
:
w h i c h :is e a s i l y s e e n n o t t o b e e q u i v a l e n , r r e
vx
~,.,,Rxv. :
: '
: ::
:
: ::
: ::
:: :
2:
'
H e n c e t h e r l e ~ s s i t v o f (¢*~).? Or somethiiW., :like :it i~ T b e o r e m :t c a n b e 9:iVem ti%" ~:ia~icat h ) e i e if w e , r e a r e d e~ : n a l i t v : i i k e . m y 0 t h e r N n a r V :}eiatio]4i 5,~)wevdr :{i~.."i:;,~ ,.as.~t{.:a}: va...~.. ~t h a p p e n s
t h d t v.quah,.v ~s m,:~,¢ .I,a.r, ~,,,.~ a,,..Je,.,~r o ..................................
3. C~msequen¢~ ol tl~ existence ¢ff automorph~nt~ Obviously.. t h e t e c h n i q u e ~sed i,n ..,%:''.,,t:,v;~," ~."~...... , n be a p p l i e d to the a×iom~~t:i=:a~io~ oroblem, f o r f u n c t i o n s o t h e r ,,.hart c h o i c e fm~c~.igns. B a r M s c ,~"I~'~n~,~* considers.: m' t h e classical case, t h e c o n s e q u e n c e s : of t h e e x i s t e n c e of a aon-1~iviaf irrvo[u'tion ......... f o r a i a n g u a g e containh~g e q u a l i t y a n d a bi)iarg ~e~u~ton. " . . . . . In fl~e intuiti0nistic case: w e n e e d n o t use so fancy., an ~?u~omorphism ~o ot~tam., a non~trivial set o f ,~x.k.,,.°s"-,,' ......, T h e e q u a l i t y conse{mences o f the e x i s t e n c e o f ~" ....... ...,,a,~a, m.m>~:g.~:~}~1'~{gm <: ~-, per:m~,ta*.~o~? :~r~; axiomaii,~ed by !~:e s c h e m e . ~ x y V x i Yi ~ y i x'¢" ' -V.x..y ~:1.;, x ,~'-~s = y ,:x...~,u{ .... ,'~ ..... ' -- ~" " ~ ' - " > - - x T - < + y{ = . e v ~ W h i l e i~: elm~sical ~oe.ic '~ -~ . . . . . . . . . .
~:,..:-.'-
3xy(-~-:,x = .v}, guarani{eel,s< . . ~he e x i s.k nee of. {wo d:is~inct . eIeme~V.s to > .......... I.,.~ rod,,,_ , ...... , a thfs is not so i:,a Luttfitio~istic 1ogle ...... as the fo[lowim, mode1 shows: K:
¢.~ } i a , ~ , c )
If K w e r e ", m o d e l of t h e s e axioms, ~he~- w e w o u M also h a x e ([ettin¢ ~, ......"- a,~d s~.~m°ess;e~,.: . . . . . . . ~ the v a r i a b l e s witl~ s~~perscript 2), ~. ~;, w,- ~v, w~; :~?:{i.,~ := v,, ,'-,,u - v )
C h o o s e 34, Y':'. for .x,, .va as..umh~g the wd>.es <,L c, r e s p e c t i v e l y . T h e n <,~:>Ik.a ,= a-.~e*o k x~ = *,=~ v, is b
cg.,ikc= ~.;-~
~v::, is b::~,-~, }-~,...... v.,. ?!~>.t
a c o n t r a d i c t i o n . ~5~. ~. ~'he aEcwe s c h e m e is n o t
3xy (°'-~x: =
ntuitk:mi,,~tfic~:~lty d e r i v a b k
3'ore
yk
~do~c generally, s n p p o s e L ~i>; ~::~ ~V~ ~ co.nta[ililla e q u M i t v ,rod '!I" a t h e o r y of
L. Tl~c L...eotlseqttel~tces Of lhe: ex:i:;(e~':K:::~~>f )~ ~Oi~r,t;-ivial homomomhi,~mare axi o m a t i z e d o v e r T by the s c h e m e a s s e ~ . ~ ; non-trivial homomorl~hisms:
..... ,.6~, exis, e n c : c~f ex:tendible 11hire
where ¢~xx:F • . : , d e n o t e s an: a m m i : c f o r m u l a w i t h v a r i a b l e s f r ~
i
~:, X~,.
x..
~{t make it on~o:
H I e c-xisDix~cc o f <'~ k } t o f :~i~to-Gt©~'sLdsms~
:ha d ! S-=='~tCi~ ' i~;~1 : l l ! p p l - U
t~Ic,,"I~II
ii sIIr~(IIUt[I is. c : l i"i e d
a 1>m~e:-~<:# I , "add~ e q c : a i i t ' , , "~:"~<~'>{ w i t h
V f .V. & ..
0"~f
b~,i:i ~'==', : ' ' , : ' F : ' , ,
.....
a~Itol~lO'~i~I}WiG'II .
'P
.
-
w < , : i I,~=~.
- "
i!)s@?:t, i i i ,.. { } i i t
"adti~ [ 8 8 , 0 f ) l ! t O t h e R/k:~-
"
"
~
~t i-J-l
d¢,mcri;-
~21iI
=~=~=_~y
±l~z~!cl=..l,~ ....... ¢" ~"0 1 1
~!Q~;It}~I i
i
,L<,-I'!IUI£1i!'I: ..'"' ............ . . . . . . . ':
L " +d, > . ..... . = t l C""
I f V:v i y ( : : = fy)
(HC)t,42)
V££ ah Vx(S.,.,; :-:-,J:)>O V:,'q+, w , v , .
!i!Y II [i
llI[~.,-iiiO',:!_-~I~Ibg]L ~.~IVi)~'I I !.ii.~)0~1' = iiI
6',,=,=,,+;,=~===v4=uP. . . . . . . ~,,~=~t:~ce . . . . ;,-~,'~..~ =111(~ 8 1 i o" i 7 ! £
~"=l;~t':w¢ 11o'41
i1 I11 W
III " {}I: i' 11) ~1IIG('O i i i
~:O 11!\,' (){:iie~' ~iV(:)[l ,.,,,,,,dL~,4~, ~-~I~
(HO>.,~3~
"
(~;O~,'~4.)
Vxy ~/(y = Ix:.
,,................. ,
',.,,doms ( H O M [)-(HOM4"-.' ~ae,~..O, " ,'-' assert tin~i ~:]}e Amct:~o:[ts " " fot'm a g;:o;:rf'~ of ~c~tO~t~O £ 1 h h ~ r i l s a~ld,_ :-s t h e y . d:o ;.~oc . . . . -vlc,,,= q--' +~.=I,~+-~,x.s" '{q'~.cnc,~'~ o f a s~c;~--ti'i"-.,~a i g ~ [ o m o r o f P
ism, arc'. cortservative over TL The official 8.xiom is the i'ans~tiv;.ty P,,x~.~.~.tLt , : , , : : : L :. W i t h it.. we can prove t h e scheme,, expressibl e .inL~ of .o(.ao~...: no,,o,.sea,..~' ................... it".>.. Sm~tences asserting t h e existence Of back-and-forth eXtendible finite " q - , - '~->, isms taking any given elemm~t x to any other give~ eleme.m: y: Vxy Vxiy:,: 3y~x7" • *Vx,.v~;~vi,:V; .
.
.
each @'X"+ .x~ atomic ~=ith variables from t~e list x . . . . , T + (Horn),
.
.
.
vi)> (~-[om}
~ xi;. ;~?e (:t6t~f.Tt© =. .... to ~?@ . . . . . . ']"~~
P r o o f . As bcfoee, w e ~.v:ll expand an appropriate mode] o i" )t',: ' ~ ""' .1{ e } ) t , ( o n { ~ o f ('F * ~. ° ~ . , {~:})foI a n y semence @. t::it'st, ~.~c Should explain tile: .mteru~ , , a t t c t i O:f ,fu.~tct~oi) "~a '+:'hI"`=',-.= ,+.,..<,.,. ~ s g; I t : p i e modetl When we had a sinide funct.ion ~onst;ant ii~ Che proo7 o~ Tl~ei:n:em 9 I v.,e gavc i t an indexed famik* of fuactio~s as its :inmrPretatiOp.o \Vitb fimcdor> vat:iaNes] We are cons~:derirt~ functions aS bei~ag ':objects : . . s(;,~,/t . . and, h;;ce ": o f .a ~~.ew other obie.¢ts, they titi~ht. O l l k ' cOrlle imo: existence at ~iddes cmite 1~:',4.~,,~,,~,~:i-~,,:.,.,~....
i
2.08
~2 .S.+,'im'~~ ~ i~
~
~
tree~ W h e n ar~ o b i e c t of this i e w sor~ c a m p s i a t o !~.~'~;i~;it :no:Ie r o w e ,a'_~.erprct i~ as a partial i n d e × e d fami!v of ~m~l, io~ls ~~ ) , ~ k~a~c :;~ ~-anges ~.,,vcr all the iiS&:s ;it wlfich the o b j e c t exists (in :pa:r{iclular 3 > a4 a n d , < h e ~ t:%:,e f..m~i>~ safi/~fie~ :Ore compatibilky condkiom
]Let K be a tree mode~ particular, w e ba~e
m,~
W e defiae n e w coIlcctio~s of ttmct~o~,:~ +u, ~sla~'~s ~v E. -=-',#<~'"<' : (e~, ~ a a C;~ "' C z Usip,~= the G S d e ! p a i r i n g ft-:nclio~, we ..';m c?!oes~.' . . . . " a~ ,.~"*m~a,~r:~e;'-'~.~,,~
at]d
{]).{is ~,e<{ L]I1 ©{~N.ITI{~F~!tlIO{! o f OIlY -qTW ,d,t~c:~c.,.~ C(),~tS[3IH.g.
E., = { f%':~"q'.~<-x , --- ~t ~*'e~ [~,
Wii
be
"die
{~mction
e~-, %. >
domaim
say
,
~o
D}a',~,
a<+
~,,.
~a~<~,;t~< o+( fx~ e~g ;}~e, COI18Klfl~:; j ;,~.~: f% ~,~..... ,v a.~; , . f!t<3 l!tle~.r'*l'J'13{ioI~ -
vr>e..n co.. we w i l !IK~uctloII 0{1 ey i'l SDC|1 a wSy as 1o g~Hff[trllCC {A,~, ~a, ,1 ,(~ ~'IK;Ip.~Dev, Oi'~e."{o.-o~].¢ alId OatO O%~ a;Kt ])r~>;e'l+YOs the cqti/:iiTty and ;El o t h e r a t o m i c rekttions i~ {~)th ,.'~;,~x:a~"~u.>o-" Fur!L-:er. we sBaii do fitis ili s u c h a way. t h a : tile: f01lo~i;~g=. ~ co~xfitions u ~ , M f o r a l l a, & c, x t--.,fX~:,, = G, : ~'" (t) ~<'""~ ' °'
........
(~) ,f 77 ,~'~(a} = 5+
:
:
[ N o t e that" we. Dave Mentitv,. at1(t !lOt e q u a t i % ,,%, written hered" C o n d i t k m s 1.a~ w i l l further .. £tx~.!r ~Ite-:~ i:}mt -+'i, -= . . . thus. t r o. t (ttOM:I)Z(HOM4"t . . t 2 ' t ' " = (]~":";'~-t and will b e valid. ( H O M 5 ~ w{;. f ' > H o w fTom c o n d i t i o n 3, : S u p p o s e at n o d e e:~,~,,¢~{,,~,< b¢.e~ defined for' all p<:,:r a n d lhai i:t is o.~xe.i'o-ofio onto, a n d l_~,eSe~:ves all alot;~7.q¢ rN:~lioas i;l iX)Itl d i r e c t i o 0 s , ix.~; :for R :a~0mi c a r i d
a',= • l-Rxo
' 'x,. <-~>a,, if- .Ri%;.(rq)"
:
:
T o define .t':~, conslder'ta,,;b,,).' Case i. a~, = b~. :'INert define i £: (x:)=.v : f o r a l l
"'=:~:" # ~ k , {.... ,A!w. ~ ,L :
x~!.JDas,,
:
:
,
:
r
: :;
:
:
dl
for
:v ~!: r)~,
cx:e::ds
.S[:,.:. T E c
:~':',~',:...... o f C a s e s
:>3
"° :'~
......
"
c o ~ ~'rs cc,::d~dc,',: ' 3 a ~ : d .:::so r:::::kc,~ c'c:zrv ~'~" . :: ..... .............~1:-,-~, ........
P ~ : ° $ V,.., %- .
::f .
.
.
c..... o : 1 < ~": t t.o : :
.
.
. .3°
.
.
f h.v "
.
.
.
.
. o:~.c,,..,:::~:: 4:ii} '..:::d :1, ford.: ~:' :~ ......... 4.i)
~?,:{~):!.qJo:: . . . t } a s 1: {)::c~: :...uocs.s,2 ~'~ "
c(::~::~:p0n O ~:I:
5~:,c:,_ " ~ " " ~ "~
$.L
.... qh,,.~ ),. : . e i
p :~:'; p =. i + 2e}. f o r s o. m e bh:ite ordirm: ;:' ::rid h::ut o r zero " ¢ v e m .ue. . . . d
',,vhcvc ~":*,'; '"
t h 6 * ' 4:'.:.:8Iv;v..~,,...,,.'~d~'~:~*'~e'W t {~,.'t. r'~C,~{:{: l l e
be
":-'*~o f :,d: for:~:u:ae w~{h
,~'S~..~'.v :1., i % , q:/ < g , ,
k.~,::~:,2s ............: -
3:rcfi:::°
~ .3 " .£, 2 .2 V.~, }:v~2 .~v:.: :" "g.G,v,. ~x i.d{,.
a n d matrix. /~
r
a ,¢
:
{
">
W:. G,,~.
.....
;e.,}
w h e r e p : , . : . . ~1,. < O a i : d e a c h @ is atomic. T h e axioms ( H 0 m ) a n d the : -,.:,'C:;..... ~::J :~:,,, :)...~ ...... ~, assump~ion tba~ each ,:;v reahzes r ! x v ) o : r cG. rea:~t~:cs '~.,,:x,~ t::. I©0 de.Eh:;c:; acct::.rdi{tg ;-~s ~,' <: p was eVef~ Or o d d implies that %tY:~ " ',* ,s ~- a type. ~ s{ ~, ~e q-~¢ fii:s¢ ctm_S:~:~m* n o ! j~l , { h e l.,ot i ~ te~%;,:,, ¢ -: ~ " ". . . . . sh-:~ce t h a : rcalizeS ~';:,o iSuch a c o :: s t a : t ex:s:s y~c~c~s lhe t x : s t c ~ y e <.~ ,X,.... o~..~:.d~a Cod>{aota :x~..al........ : g ~ ';.) &d;casc :4/i" p is odd.: ;..e~ G b e t h e first c 0 n s t : m t :~oi:i s :the Iist !¢c..,,,;,..<;,:-ar, d •
,
,
.
clcfine: 2 ) • 'r,~:(X:} ='LI-
~2' ; -'" . , : ; ~ ;~, 0~,
:
:
: :
!"~b~'): consistin,a Of all f o m m D e as in ~. b u t widi d:e m.atr[ces, :
:
: ::
21C
C. 5'~<~r}':;xk;
p > , . . ~% . ~, .<.. p. ,~ e::tb~ . . .r ;. iS . .a .t3~e .
list
p-d.,,
we c~a~ ~'ealize it i_~va co,~~stm~t J , ~ot i~ the
.td~,.~,<~,.
T h e constructions, o f C a s e 4 is comp*ee*~d by ~,~{q~,=om-, #:~ ¢,4 "~= .... for air # < S~,_~.~,o
] ~ e Iiew m o d e l , say K H°'`~ is o b t a h l e d f~'em K by addir~'~, d'~e ~e~w d o m a i ~ s 1 D K ~ ~ = F,~ a n d i n t e r p r e f i > g f ~ b y the fam:!v }:~÷> ........
: T h e b a c k - a n d - f o r t h c o n s m a c t i o n of C a s e 4 mm:~ar~tees .t~mt iis hmcr.:i0nS ~[[ m a p Dc~,,. o n e - t o - ( m e o n t o itself. Cases t - 3 p r e s e r v e d:is p w p e r t y . M o r e o v e r , ~:he ....... ~?~.~.>~" r% r- g u a r a n t e e t h a t
for all a t o m i c R, aIt ,a~. . . . . o:oiF V f g x t . . . . . . . .
~;;,,~~ CIr, a ~ d a[~ :~4~propriaie f:'. T h u s . ""
.
~- . . . .
a e d ( H O M i ) is valN~ ( H E M 2 ) foil:~ws f r o m d~e fact tlmt e a c h .{~, is o a i o . v . > O M a ) - v H O M 4 ) tohow h°om the tream~c~~t of' C a s e s I-3~ Fir,,a!b< :{H-]M~% fottows p r i m a r i l y f r o m Case 4. L e t ~ i:~e mh~i:mal sucia ~]~a~ a, 5 ~ C ; . . ~,,v g,',
b ' fali into a case o t h e r thm~ 4, t~*se condifiot~s I - 2 .,:~d,~i n d u c t i o m This complete:s {h~ proof° qoexl.
T ~ e examp,c', ~'~ of tile ~',ve~lI]h.{l~," ~; el'e {hi~-; sectio~t si~owg t h a t F ( ]~<'':>~'~ is ~tot co~servative o v e r E Q (the t h e e O of equa{ky)o h!de:ed~ Li)ch#z i13] ( a n d ~ater the a u t h o r - - - c ~ . ~,o~ c o n v l n c b g l y demonst:rate~S the :~or~>-homoge~mib' of e(pm~ii:y by
V y ( x = y v '7:v = y). It m i g h t have o c c m a e d ~o the r e a d e r t h a t we could have prove:~ T h e o r e m [ m o t e easily by t r e a t i n g all ,of o u r function,s as izr~ C a s e 4 at~d thews s i m p l y c i o s ~ ( : u n d e r comoositio:t~ arid ~.v 7. ~e--" _,._,:qon, I n d e e d , we c:otfld h a v e dm~e this, O m ° 'prooL in aa~isiyi~ag conditio~as i - 3 , wa~ ar:: a t t e m p t to m a k e {&e growp of autom
SY :'.
(UNtC)
~i'~'~. C o n s i d e r how w~c w:>,,~h:i a;t~.m::~p~ to d o this. O b v i o ~ s l y , ~ e w o u l d w;m~ t:o sati-4y a ~ew c o ~ d i l % n : Fo~ a:!{ ~,,, }< c. ~;~ s: < / > a 4, ~'*"~'~(c~= d ..... P"~'>{a:} = e':, <'~,.~ Actuallv, this is m)t e t , o ~ h : : W e woukt ~.......
....
H o w could we d o this? W e d o ~ot even h a v e m conside:r w h a t n e w ii,~t,S~.:to a d d to spot the difficulty: t~i o u r t r e a t m e m al×~ve of C a s e s I.~.4 handI{~,g coad{!~ons 1.,~!~
21!
"~ " ' r " s ; o f h m c ¢ i o ~ s , i t ..... t~**?:80~t lJ~a{~ _ .... s e ( U N i C ) o e o a r Lm,,m,.:,. , " : . IO imp,o, FItUS{ ]i[{IVe e i ' c ~ a t e t :
be,;~lD?~tV
.... ,,. ~* ~
"
~.r*mM }. ~l['.x ~¢~'
h,mctioi:s~ w e
i~l {.~le l~'Oe:'~s'I q ' ~ # d p~(.,-.~,;,,{~a *.;:Gt,,
L,,,.a .......... , . . , . l e t ~s .... b.............. " .... re~- ":~:,o<'o:F 'r" ~-,, t h e a<~,L.w.,,oze ~.a <.~.2{x¢Oa~i?.
oi>eral k)tl, I r,-, :aotc t h a t a ce.~,'.se~'aue.uce o f ( U N : [ C ) is
t:;'x) ! =-g " - i z ) , and exte~siomdity
i~ a l l t h r e e v a r i a b l e s ,
OLV' g r o u p o p e r ~ ; t i o a is o b t a i n e d b y c h o o s i n g a n e l e m e n t e t o b e t h e k ~ e n t i t y a n d co-~.~m~erm~ : A"~,-<" ,, ~-, Oti Lt}@ i e f t by... iV: . . . . e a c h f m v c t i o n .ia:~.~ t o b c- .~l~L_uel~cah,,~ X " y =,
~<¢ x} "~,'~ ._ g(*, xM' **(e -r"-.--"~ x',
~,~J .-..j . . . . . . ~{ . . . . . . ~v;JAa
'~Ve, h a v e :
x , ( y , z)--,-~,~ " - . y;" -~.V-~ ~, is j u s t the; e x t e n s i o n a t i t y e-
,*: = f<:"'"~Cx:) = x .
x"
e =/u:'=>(e)
(04) o f F. F o r (G;:,,, o b s e r v e
m's.: t h a t
Also = ,x.
F o r i(~:~,, n o t e t h a L s i n c e f!<~::' i s a,~ a u t o m o _ ~ p h i s m , '4~ere is s;om e v s~,?ch !.ha* t m " ) ( v ) ,=: e. T i r o s X . y =~ e. B i , t t!~is m e a n s of"(t:,'.)g(e.Y)¢i~ ~ g ~ ¢ @ - e ) f p ' t
(UNIC)
wheace
f '=(t'"
yields )
and
y.x=e,
e#.~.
F i n a l l y , w e v e r i f y ivy4). N o t i c e
2
,
?
{.*~,~.~.L4. . . .
a : { e r , F ~ ' k v :!.." . ~ ,
~'"
~¢e)=:~.e-ay =-]
........ U
...... i c'~,
whence t .... ~ x D = gv ..1, e {,<.~ .'.
fh'~.a.s. ( G i)--(C14) a r e ckr;T~:abk> ..... L,~ "~"~>~l ~. . .kkNtv" . ""/ f o r t h e ~:<.,,,:~,.~~:~a ,=m~{dp[icatkm~
Note, '" +~ :i~.rmer " ~* e~,.o~,'.,a,~,vthe .e~' . .oU{} . . oDe-.':a)~i,o~a is ...e-*" ~,~a~h=,%w ~,,.#~-,'..:, ~.,.o.,,~..
1~©r, x . x~ ~: .e k. ~ a~,,d ;v.,-, is alt k~H~O~!k?i-g;~'ilSP-1, t_et L "t.,c" e. ?',a..~e~l.t~,.~3,."--~ " , ~ " as *~-+:"~..v~c .~, (};d {et :, be ;~ n e w c o , . i s [ a m . , a ~lgw. Dirmrv. ~?i:~erar.io~. s-vmbeio -::-,.~id~:c ,_,,,,~..,..-w ~x-~-,mly fvme'ior~ cc :~.sta~t. % ' e ~-~...,+,~.~ " defi',~.e 'theories" °i{ • "c>~ a',~d T ~: i~ ~.he ? a ~ a ~ a g e s ~~ , .J. . .ie. °} av, d L U {F}, . *-"~ ..¢..>) ........ . . . . .:.' .".' : "~. 'I *:~ exte*ads T b y the axioms (GII-(G4)a~d ({.l~,:v] ~::,s>,;= . . . . . . ~ m g th;.~i . {s a !eft h w a r i ~ m t o,-om~ .
e ~m
~,,~ ~
.
.
.
.
.
.
.
I
t h e s e var{a~qc.s. T h e a x i o m s k~ ..e
T I
" ~'v,. N,IC.~
wifb,o m
YtmcIx),i
ll'C:
(Fi)
V:vy:v ~Tiw(F.vy~.... Vx)'e wilF~e:.~' =
g)
.~'. Fx3'a'}]
(F-a)
V x y - T w ( F : v y a = w-..~ Fyx~v = 2 )
~t::S)
Vxy(Fxyx
=
F(y,
(th;) (FT)
y)
g x ~ y ~ x e y : : : z w ( F x ~ y ~ : = g r x y : : ....... F,,',h )'t~.v = Fx:: ),:Oe).
~"~"~ "Av"~. t~:-'°- >:.:~.~>"~ ~~ . . . . . . a:: ~:,~.q o.'°.'" f o r m a grcnm, o f a u t o m o r p h i s m s : 0 I % ) a s s e r t s f h c t.rals}tivitv .~f this ~~"~ a-, a.~{ ::i: ,.~:..~ ; , r - , . . ~ is tt~e t m i c i t v a x i o m . .Tcv~" is i ~ t e r p r e t a b t e i..'.~ '7:: A{(-.~~ 'we: vtdd a conslai~t e b,, t h e dcfit~i~.io,,~ :go ~-:: . . . . . . ..< ,. . . . .= F~e~:y. C c m v e r s c i y . T v ~s i,te,'p.~ :~ A:Ac ~~ t .... by: d e ! r u i n g b x y ; := v~" "~.a ,.W no{i!}g that ttle: u~TiGue,~ess ~?~o o f fo~ t h e i~., ~:>~o :,~,:{p,:,( ftio tis!ic e l m file irwcr'se fmi~;~k)~, t h e s a m e L-oco~iseque~ces°
: :
:
R ~ w a r d s t a t i n g an a x i o m s c h e n i e o f u n i f o r m homo~:~'~eiw, w e L ~ r o d u c e s o m e n o t a t i o n f o r s e q a e ~ i c e s of Va,'iabtes. F o ~ Z e i t h e r x or v. we.. icl z. ~ ; z{: :det(ote "
~"
~~
" ~ " a~d
z~:~, . . . .
ai~,~ r m p e c t ~ v e i y .
The
sehe~iierl ( U H c , m)
2~3 v,Tmsl;sls e{ ~dl s e ~ t C t x > ;
%¢ .-,£,A , 2 ~
\ v i l h ,- ',~';%:,,, :i**:~
.....
2-,~,
_ ~.
., ~
.,2
:{
"~
.,2
<-e~:-~,./[y~.,)..... ~> o
. , -'" y~.. ~ ':c~,: ::: . . . .
, "i:,,,,.j i ~ ....
(h)
"~, ' :": "~ '.,~.~'< ....... - v., ~:~"= x~,., *
~?,, {?* ~'8.~a£{1"V.{ OVQF ~ ~ "q
:= ":'?P~L
(:.,~.,
}~',A
~ . [ C q ::=:.v~ A ~. . . . xi '~ 5'e~ =: )
,,)
.,,.
,
,
->/H\UvL =: e(." a. v¢ =
.
. . . .
""
,,' • ,,~'"~
.,7.s
~he ...... ~ " ~ e~'m , ..... gU) . {s . t h. e .uniIb~7I~itv.. . .eorm,p.~: . . . ................ .'~~ t - £ {He app:ro:~qmab.on" ° t o <,~]..,~.),/-r j ->~ ",'-~ L<:." ( F 7 X ~md " . . . . . . . ,. T ~ w o , ." d e i ~ o t e r + ( U H . o m ) . co:mpositR>n acc,?~Rm~., ~'"~. t o (,_m~x , r , . , . -vv. g , c ~~<.t {:~ g . e a c r a L m*'~'~<~':"* is a ~ r o p e v e x t e t ~ s i o a o~ 'I "~-:~c'~,~ w i t h r e s g e c ; t o t h e ~ c o u s e q - u e a c , : % T h } s wit} f o l l o w r a t h e ~ ........ <.-.,-,~. . .•. . . . . . . + ' ~ ..~ w i t h r~...~.., ~,-,,.'+. t o . t h e. - . - . oi a ci~.ca<;,d.,,g, e~>.~'ai{tv.
H i n l to t h e proof.: E x p a n d
a s u h a N . e ~,< .......,_,<~* o*{ ("i'u>~°'% {q=,}) to e.n,:: c4 (°F ~ ~{.q.'L Defir}e .F b y ,:he us~m[ i n d u c t i 0 ~ . T h e ,':. , a c k' ~- a t d ~ , o...... , . , .p r e ~ x a~iow7 (}73) t o b e s a d s i i e d ; (h~ .... a c c o u m : s f o r ( F 2 ) . (F6'i :,,. (u) a c c 0 t m t s g-,o~,-(F:I), *iF-:':}, a n d (c~. accoul~ts {'or ( F 4 ) . (F5} is r e d u n d a n t . Ge.d. Reversio~a the multiplication ~ th
right t r a n s l a t i o n
g i v e s a corrcspe,,:~dis~ r e s ~ i t fc;,: T ~sR ~b*m,, a ~ . i
invariance
in: p l a t ©
~.~f } e f t
~;f~.~sIati0n
invar;i~:~nce. ~
L,
c e n t a i n ~ ; o n J v e q u a f i t v J w e h a v e b o t h ri.gi~t arid l e f t trat~slatfor~ i n v a ~ i a n c e ..... b v t h e e x ~ e n s i o n a ! i t y o f t h e g r o u p o p e r a t i o n , l~n ~..,eneraL d u a l hw~Lrianee ca~m.ot <~, ~:~;~,,~:~,, be. imF~)sed. ] b i m p o s e d n a i im, a r i a ~ c e whe~~ L c o m a i n s a~. a p a r ~ a e ~ s r e i a d o n as w N I ae; equa{ilv~ R~p a c e ~;qu.:3iitv b y m:~aet,~es:s {e (F*'~ ( P 7 5 > 4 ~, ~ m d (~:~ < S i s x~mald :~t~t~. b e n e c e s s a r y if ~he grcn?R o p e r a ~ c r ~ w e r e abe{ia~% :S~,r,,,,ev~ ~ ,,,,-e .........~.
C . ~ n ~ e ~ e ~ a m p l e ; i'l"~e h~tr~itkmis{ic t h e o r y A G o f abe{~a~ g r o u p s iS r~o~ coa.sei:~ at:ire o v e r t h e i m t uhi ~ ieo:noi s tri c y : ~. G o f g r o u p s w i t h :respec, t ,:o ,,~-:-'..... .~>;.,...~ % , ~ w~.. >. a x i o m a t i z e A G b y m e a n s 0 f_ t#G"[~. - 7 t"£~~ 4'=" )alld : : i i: :
X':y~:y''~
. :.
'
{
:
(A)
214
(z s m o r ~ a @ i
;
If we consider the functions f~z =: x" z a~d/;.~ ::: v ~ z . then x,e get £ y = k,x,
f ~ e =: x,
L e ~= v,
=
.
whence ";~,'~,,-,': ,~; t h e :fir~e poims at~31I{ b~ck--:md~orth cxwndibD e~efixes'} we~ cai~ derive every sentence ~ith .a pre.f}*,
and ma :fix eo,asisfine, or" coa:hmc:~kms, of the tt.%~',~m~.L
•
yi
.....
aI~G~
where e agai,.~ ranges over {t, 2}. In the frace of (Horn), the cxis~emiat q~antifier. ~e, can be replaced by a m~ive.~at one---~ wh:cncc, i~ G, we c:m assume t[~e variaNe to be realized by fl~e grxmp idm~ti b' e. ~INe ciassical rlon-abelia:~ di{~ed:ra~ group D:~ a~d two of its quotients afford us a simple model of G in whici~ one of these theorems of A G fails to be forced: K: ..... f
Suppose the sentences i~1 question were: tree ia K fo,, a, v ;~s:;un,i;lg the values a ,es~:~ecfivelw We consider several choices of .v~ a,~d fi~e co,fresh,old{m- va{ues o i ~,!,
.
=e-..
a d . ,~
-- a...
,'~z,, u o- a:::> ~ ~7: 9:~,
a& a b : ~ } .
F r o m these catemat,ons we ~mn conclude that u is ab: U s i n e tt,ffs value of **, we shall make a few ~urther calculations aBd hi{: u~xm an in¢onsi?.~<':>cv~ X) :=: dO 2
_
~
u - - > a ~ t } h = u : S y ~ is b.
=
. . . . £
}1
:
:
:
:
: ;fOg' l.~t/~ 9 7 7 t g ~t~%
,:v ~.t:.v!{
.
.
.
.
.
.
,!,~g,a Y~!,
....
<::,>ii-:,:~ :-~ x:{<-{,V~ . . . . :: ~(gg) !;°' f ~ O
ca:
, ~-
: : : d}~
m s / i x ] i c t i ~ e Vhe fact thai v,0 ~ ¢~.~,, \Ve ~~..... ,-,~o-s........ ~-,' ~ ~ s u i t a b l e t:heorv T A° <.x~v.~ c~_**Iga ~wen theo>.~ T, ~:.Ve ,a..,~c t~aI \-<"k' b e c o m e s
kQ~xy(F~:~:xy-'= 7he,x)
:
i' >(%7 .,-Q
w..,~ . . . . nC,R._hs .F, 0 umvers~} .... eP; and r i g h t ma~tip~k:a/ior~, .~.::~.,,.,..,.,,;,"~'~ .......~;";'~* a~'td con.'qder a s c h e m e of b i - t m i f o r m h o m o g e n e i t y .
4. ~ m e ~nmsequence~ t o r a decidable e.q--l.~IRy A s ~tm~sed b y BarwTse, l~e a x i o m Schemat-a o n e gives in results K e v e r ftmctk'm is being._, cot~sidered. T h e form, of +*'-,,~ea x i o m s , ~"~,~o,.~--'wkD thelr' ne cessity c a a lye v i e w e d as a~ indicatio',,~ a n d function of expressive power. TEe .greater expressiver~ers of the inmitio~fistic, ianguage~, of e q u a l i t y o v e r ~ts c.:~.,:,.,.~ ~ " w..... ~ c o u n t e r p a r t tncnifestS itself in the n e e d D r such a p p r o x i m a t i o l s . T h e "-*~,_~,~aH,:,~,t~S ,'~"~~',J~.~A~eow,,of decMab!c equality, h o w e v e r , i~ not v e r y e x p r e s s i v e ar,.d, wne, r:-o a d d > *kmat a x i o m s are n.eeded, t h e y are. r e a s o n a b l y simple. h~ the p r e s e u c e Of d e c i d a b i l i t y , we c a n restrict Our at tent{oP, ~-o :-'~oi'~.~o.~ m o a e l s ........i,e; those ii~ wb.ich e q u a I i t y is interpre:fed by: ide~itStv. )7o;, g~ver, a x - ~o • v.......... X g ~ . ~ t a .~,~,,,,,A~.>,; ....t.?~...q, K r i p k e m o d e l in w h i c h e q u a l i t y is d e c i d a b l e - - a n d in ~a.E~:::~; hokls .-:d (:tie C:m s i m p l y r e p l a c e t e e elemer~ts c v their e q a i v a l e n c e ClaSses w;th r e s p e c f to equality.: T h e d e c i d a b i ! i t v of o,-,ai
aild
exteilSlonhllllVthat the: trtlth rabies (i.~, O:f t h e Statements a 1- % a Ikt ~::>)do ~(H
(lii'Ae:Se~ o~1~: ~,an~s ol:,lv o~!e CO}lSta[it' p e r O b j e c t ) ; : t h e Stn:,.Ctm. q ,e.w~,,~):; o r ti':~e forcing r e l a t i o n . In. p a r t i c u l a r ; if o n e starts wit2; a m o d e l : K b~i~e or~ a tree, :ihe :~:i:>::Y~<~,,Koeedure will vie,ld a n o r m a l i n o d e l K N: buil:t ,~m the s a m e tree. { i
:
~!16
C: S~:~'~.~ ~f
L e t (D'..', deno>~: {~.....
~
......; - • ":~'
" g%'¥~
V x y @ : y v - ~ x := v?.-. .As m e n t i o n e d
~,.~a
m. Sect~e,.~ 2, {f T is ~ 0:.eo=:v ¢o~i~amm¢ a b ~ ' ~ r ~
re=:<~on ~y}l£?Oi
R a n d equali%~, a n d '~,; ~ I-e is t h e ex~e;.'_s~o.s . . o f . "P .h y .: ~ e a d d : i t i o a " r -, eonsim'~t ,f a~~d t h e c'Loice a:.~d f u n c t i o n a £ t v ax~.,,.:~ -:........~ ~,.e.,~,..; 'tLe~a: ] ~ ,D~.o:,TI
"
~s co,~serv,:~!~ve o,,e~" T + w x
.l'.,.-
f~Inclion ........V x ! v / ' & ~. ""~,~eo~c~x,~ .........
~
.'~yR'xyl
C O ~.gQCi~.tOI~CC{Q 0 f {'i*e ()xiS{:c:~lc:c
:-:uti¢:0s,. . .
',f
9. . . .~. . . .is <,:'"~,,~,=.,, !>3' ',,.'<'~rk(~< o n e ' s ,x"~x. u p ;.'~ a o r m a i
i a d u c t b : e i y A~ ~ : .....
.
eXiCilS{O~;'~; *_;:hO{Cg
0,!~
A. si*,:~plc- tro~:ff ''~' .,, ~,a,>, ;" indep~;~&x!~ o f 11_m~ (?..f
~" -~a.< K /2
w i t h .x,',-" ~-. I O i'--',l~g~)£C:{"
ire{; m o d e l
K aad
" ;~ {"IIi!C17i(HI ~, -'O~l~Zg:,~"~ j[
T. h e r e a d e. r ',~.",-'~vec~£rries . ()'-e~ . s u c h a constl'ti(t{ot] wili n o , i c e {i..~[~[ ¢~'e'r:" . . . . . . {s !~Odilk{~.¢'*" problematic
abort!
~{ at ;tli ...... ~hs ~, <)(:it~:Ii~iv ;~.s [(~c~;~{t\ i4¢c~t,' . .t:.c, . . .~]1~v . . c}m~c,~/:r':", ,,
ca~ase:d b y t h e ~4eeess{Vv o f scttisfx.'in,_l exter~s~o':~a~i-b:. A n o.',.?ier . . . . . .~ec~>=_~,.,.. . ,'~e~- ~*. ~rpp{ies . to. s h o w. ;:he {loI'~lo~e~ie{tv~ . . r%.,. & ~ e c t d a b i e e q u s ! ~ v . Le~ 1DEQ~ t i l e Ib.soCv o f a ¢,~=,..~.~,a.,ve .-~...: a,.~.~ :. eq,~a~ity, ex~e~:d {he ~b.eorv . E Q . or" equa.Ht3 b}' ¢~e ,,.=...,<.~'~¢~,=;*' ~ ;oi2 a x i o m (DI-. g ..... ~, DEQ
........ is CO~I~SI!'%~iltl~ :() O)'ie?i" D E e
T<" :"='*'='~eVhiS= v~:m ~iv acv~'~: IL~at i :q~ ( v u t o . m o m h l s n ! o f n ,]ocr:13}, m o d e } K o f ~~[-:O iS ;'~+ =~.~,,, a n i n i e x e d
! a m i l v 1~: I
- e l" pe'a~.ut.qtk'ms o{" ~he i~dividu:.:!.I u@m"% ._..~,..~s O a
i n d ii. i~,' (sat{sii=~nv ~nc "' ns~.~:I{ i>.c ' ". .{~Sl~io>!. ... "
!~L
~iI} = (&
]~:: r~.~ ~, i h c (,~' (-',.-'ilS.I3( )S I' i"i O
1i
fs/n{}y
x:::::b
[
6:v
~£
.v={e
is k~sI s u c h a 4. o, :'~ s so.g'i o{ a~'L*?31~glv:l't{ "~" et' .,. ~. ( ( .~S' FiO[ WOi'k 4,.,,,~ cOlisi('~ci'il]'> i~ L"['O~,IDo~)el'atio~ .,~,) . . . . . .,~ . . .::,,v,;,:,~:, .... G {' (|i)I =: t)F:;,Q ':~:~"a n d A:G + ( D ) ~: D E O A 0 ~:
-~ e. . . . ~{ w~=: dclmc'~' t!*e
s c n t m ~ c e s (.e > 2),
E } , : ~ x c " ":r...(~:,~. x:~ e x } t ,
......
,
t h e m iu D E e e v e r y s ~ n t e n c e is e q u i v a t e n ~ to a p r o ~ x ) s i t i o n il ~x'i,--/~q i a t i o n o f E "s o. r. .o.u I:)s a v e ¢ 0 n c e m e d , t h e O b v i o u s . c a r d i n a l i t V# . r~.su ....... l. "is L a e.C x,~here/, ~. . . . a. n ~ e : s
£heo~em:
[fG
i:s.a s u b g ; o u ~
~... ,~:~fi~t,:',.,,. :o.,-q,~.~...r,.,,.-}£ th.~-~n t b c orde:r' o f C- d~vid,:~ '~hat ....
o[ **
W e def}-v,e ax[orus.
e
"
"
..-Fkk 44~-Z).
LqC~£~S~O~'H
From
.
.
.
.
m
t
~,
~
-[
\
•
.q
.
l.,i
~"
"
U,*4-~ i t e a s u v i'o[Io> s t h a t (.~ + kZO r (~.mg). ~ o r . assuage iiC t o b e a h , o a ~ o.0 {.;
~. ~:' ..... >~ ~. , . .~:., h
'~ L > ~
...... > He~ce
we
have
~}[1~.e~ f o r 8O~A:~,6
~e,.°*'o'"~.,...~-s (&,,,
'I'hco~'em, H e n c e we have s h o v n G + t.E ~ to e.xtenc~ D E Q + {Lag), Nc, ,, ~.om].~ , a t mode! K ot" D E Q satisfies (Lag) if~ K sa:,sfi~.. , ~",'xt, note that ,:~. ~ *: .... ~: v~ f3 m~4 D m D~:~ iiuite =;; card i.b,7.., i carc~ tDFb'
( '::
Half ~f this we ah'eadv, know: Otw p r o o f that G +-(D) k (Lag)~ c,epe~eea . . . . only on the satisfaction of ( * ). F o r tiie com,~erse., suppose K is:a normal m o d e l of v.~ (* a~¢~,. a n d that ¢~< ~ wifl:~ Oe~, D~tl h a v i n g cardinatities ,'~ < m. T h e n
yiekl a%/~:,,~-.-~ ~.,.o.,,.,,> B u t ~ Since the cmdi.~m~ity o f D>:> ~:s m,~;t~(m.,n)~, we have m = ~(tn, n), i.e. ,,! ..... i.e. ( * ) h o l d s , a0 ' t ~ complete the :proof, let K t m the saturated tre~.: m o d e i o~ the v>_m~ of T h e o r e m t.l~ Wtien we .norma!bie, the cardh,Nities ~' d.onm kv~s ca~, ~--'~oc~,.~.:~,.,,;., .... b a t not that nltlch: T h e folhywillg holds: "
-%-~e
e < ¢.~ and D a h a s iW]nite cardinal ,~: ..,~Dp :[
[Note:we
.
.
.
.
:
c o u l d as e a s t' °D c ; ~<*~d . . . . . . : . . . . .]'i}.~;X)~Itl .. ~°*
~ . . - L ~.~
La~, v o , , c h m , 4 t a , :
:
. . . .
a,~t., "1 , :'.t: & ~ l' i d' "gi~u . . . . . . . .~,b'[I~,,,a~th.~Sa ...... <..., .............. A
,J,.,,.h~,.a.=,5,~ ......... },~.a~,
mode.ls to satisfy: (* * ) , ] It iS flOW a trh@d mattc:r t0 define abelian ~7 £r("V} opexatio~S on t l l e d o m a i n s D a inductively uP the t r e e : G i v e D~.-'.:o a n y abelia~~ g r o u p structure a n d proceed f r o m t h e r e - - a n y : abelian: ~ a o u p 0s~ L,+'Cg ~" C~t,t " ~ 'k)c, .... extended t o an abellai:~ g'ro~ip. 0~:~ L)i3 ~ :-:: ":' ' :~ *" "" *~"" ; " ...... ,..,..:.Ax
2t8
G Sme;ynski
. '-~-~ . orde~-~::d . system. ..... Of .~:rol]es. is condition ( * ) or ,( * * ) holds~ " , [~c - ~esu~t:t~.~ .... •. !:vart~d~v eas~ty seen to tm a Kripke model of A G + ( D ) . q.e°d. 1 ~ e ~ r e m 2. D E Q + (]~tg) is decidab~co Ratl',.er than prove T h e o r e m 2: we refer the r e a d m ,a m~ ~ pgu::,f of the decidability. of D E Q m ' ~e] ~-i fo:r a~ example of t|~e teclmiqtic ~o ~ nsed ........~t'fich is to reduce the decision ppc~lflem for D E Q + (Lag} via Lifschitz' quaafiiier etimina-. tion for D E Q to that of a prog~sitionN caicuh~s whose &>cidabitlty is easiJy estaNished. O b v l o ~ i y , similar resN~:s a~e obtains.able for oilier algebr~qc :~bjects~
Exan,~ !. "1-.he theoLv of s e m i g t o u ~ with decidable equaiiiy is co~servat~ve over DEQo 2. The ~&e:odes of rings, co~mnutative rmgs, eommuta:ive ring,s with utlJt eleme,~ts ~ all with decidaNe equalities--, have equality frvtgmen~,~ axiomafized by D E Q + (Lag), 3. Tl~e eql~ality consequences of the theories of inlcgraI do nains and iields wi~h deeidab!e eqtm!ities a:e axiomatized over D E Q by the sd~eme
where p i.s prime, k > ( L m > p ~ , and 0(m, iP) is the :|east p a w e r ~f g,> .m. T h e theory :is decidable~ M o ~ mundane exam)ples are given by considering the intuitionis~ic theories of fiehN of stmeified non-zero characteristi~.'s wiIh d e c | d a N e eqt;alitk~s. Even c'~assicaiiy one must add the axioms asserting that atf finite mo, l:~:!s have tile p,:o|'wr e~?:;'d~.ities: I ¢ p is the charg~c~eristic, the axioms arc
m arbLtrary as~;~ ~:, ~{s aL-<,ve. " I ~ e are the same axioms one needs to add to D E O to c:btair, the con~.q~;e~::,> ~:~:"~he ii~tuitio~istic tileories w'i~h decidable equa~it~es~
i~ this fi~]a| sectio,~, we wish to discuss various matt:crs a r i d cite an occasional co,pen" p r e b m m when it seems appropriate. We b e d n with a tech,licaI ma,{er~ T h e technique used in this paper is fairly general a n d hence ~s:~~e~y~q~-": ~,~ n~=t~a,wa}
considered the problen:~ at hand, A
:
c~as;~;;;:~:.:
classical .L-theory with o~Iy ~N~ite models~, tlaen the theory .r~-'~ e,~.e~d~,-~eT k,V a non-trivial L-aummorphis:m is ¢o,.~se~sdve ever T~ i ~ other words,, T :i'~e~:!: axiomatizes fl~e L-c'on,~eqttenc~s oa. *~ ,pad,t: Usiae the general ~-{e.~v:-<,f "- ° the autommphism by. {is ~-~p~rfox;ll,m:;~,~i~r., ~.,, .~ .,-,- woa.dd :,:~.,~f.:~L4tiiae axiO~:a?s c i t e d ;h-~ Sectioit 3 axioms :which, b y fhe way.. are still .~.~ec-:>ssarv~,: . . . . . . . . . Tile question arises: How necessary ,~,',:>°~" " ~ " .. . ~,~..,,.~. We have ix:en carehfl to demonsh°ate v; c,a~i~ythe tlee{:ssJly of the raew a:dOmatizations otlered ........that ~s; we have show~ by means of c:oumerexamp!es tl~a~ ~:~m:e new axioms were indeed needed whei~ we caffered them° Were a~.~ o~i '&era -
E~ma~ple. E Q ~L is not finitely axiomatized over EQ. gupF~ase it were. ~q ~_~e~ the equalRy fg~gment of G + ( D ) over D E Q wo,.fld ~ finitely axiomatizedo B u t it ~s clear the (Lag} is I~ot equivNent to any finite set of its if~stances. F\~r its non-finite axiomatizability prcofs that the axNm~at~afions offered fail after subjected to vario~s simplifications~ Eog. one can call for a proof that the composition mechanism (¢) ,~: .... (UHom) i s not redtmdant, or that ~:h~ scheme g no longer suflident .if the backand4orth extendiNlity prefix is replaced by the ordinaD, extendibility pre~× (i°eo with the variables x~, ?*~ deleted). One could evea ask for prooG ~::hat m~y axiomatization must have sentence~ of arbitrarily high ,alternating quantifier complexity. Another techrfical questkm: How genenfl is the method used? in the "~' ~:"";"'~ .... case, i t is quite generah Barwi~e [ t ] shows ~ha~ ' o n e ea~. get a s~quem~ ~ of appmximatkms ~?Momati~Jng the consequences, of a~y a.~v~-sentence, t;~t~.:~itirn'~sdcaiN. there is q u i t e a difference between relations and NnCtions; and ~0 ~ : ........... the quantifiers ~R and ill. Perhapa we shouk! mentio~ otu p..x~f, m OA..~. of tl~::~ theorem of van Dalen and Statman [2] o n the axiomatization of t h e e(tuaiitv fragment of t h e theory of an apartness relatiom :In this pro3f we still ~sed a saturation property, b u t not that used in t h e present paper. IX.~s that h~dicate anything? is there a difference between axiomatizing the.' eonseq~.em~o, " ~'~'; of~sentences NR~(R) :and N N @ ' ? A r e :there mrrcsponding notk~ns of res~det;~de~:~ Kri~ke m~!els!? o u r r~referenee f o r the ,'o~,erete has ke~t ,i~: from StU@hie t~J,,.; question, Perhaps the answer woatd be Useful. i : : '
b y 13a:~wlse ~o n e ~s usuath'~ intc:res~:ed in c n t y c o u m a b i ~.......... ....... t),[.cs,-~~" a• ~l~e.,~ ~ a: v . u"~ " I n fact. he' finds t h a t oiie ofte~ ~t~¢ds o ~ : v ~o x~a~iz.¢, £~?c r~:c>..rsi~, ~\~,e~ \\-'~ !~ave used t m c o u n t a b i e m o d e l s for conveme,~c:e...... we ,.,........... ,~,~. u s e d com'..t:,b~e rood*., e[s a n d o n l y c o u u t a b N m a n y t:d?es It is no,: <~:a~ ~o~ e',.c~, taat we c o ' d d ae~ by with o n l y recursive s a m r a t i o ~ : T h e recm'sio.>t!:eo:'et~,c cotmte~>a~t l:o '~ .......... .... *{{..C, ~"" is a n i~crease in c o m p l e x i % F o r . if we are ~o realize t W e s me~tioe, in,~t decisions m a d e at earlier n o d e s , we are w o r k i n g rec~.?:stvdv b ~ h e &~cMcm~.; m a d e earuer. q~,,. ra:ises q,aest~o~s: D o we n e e d this cew'..pivt;ih~:? ~g ~'.oL.w ¢ cord<{ avi>id i{:c K ' C a~dd t h e s a ~ d a r d c.:>.~:structk~1 of sa~n~,<rtend? It! ,i'7~,.>we r e m a r k e d ¢,'lat mosi of oI.u~ sc:t4hcorctii: ,)~'oo~s COIIII~[ Cl~!'}~ . ~ t ~e k ~ . . .~ . . mam~,ed . ' : ...... i~t ' P e a n o a r i t h m e t i c aug:mc~:~cd by c o n s i s t e n c y stazem e n t s and. hence.. "'-='m~_1!, resutts were construcdveiv, p r o v a b l e . K it is ~:~:eccss:rcy t h a t ~he complexity l"l l c"r e 3 s e"s ~ stK:fl' :.i ~,r:2d~::;fer ,.,>~..:, ;,~'~ d~ w m ~ M s e e m t e r c ~ n m~ e ~: " ~l!oi"e liberal
notion
o f :oF-.si:ructive ~ r o v a b i ~ i i v .
obt~i~
but ~
"{'~e s a m e b o i d s o:[ \V~,h~siei~fs i'esu}.ts ~- -' '
.
.
.
.
.
~. ys;"
"
a~rcadv c - . , : u n n l t e d t o a Igtore ~It~cr; ~ m~.>iatheor'v. . . . . ¢ tb.e rea~s or snc\v ~ to be nou-a~omafir.abIe~ tn O A F we" s ~ e s , . ' e d tha! m o r e expe:dence in a x i ) n a t i z i n ~ the c o n s e q u e n c e s of various theories might give: us a c,cd~! perst'.egth~{~ oi~ the i[<:*" -~l~,:,~<,tt:.!l.\ v e }I :we m o r e exa! Ip!eG b u t .ci;tmlof dain~_ tch:we ,~ ~>~+.....p.er~oective: g h e j u m p s ~"I-v d-v,~- *,-~ " ~ 1.~i.,e 0 t v . ~ c ~...... l e h "-.2" ~, I l I I i [ ' O ~ T ~ liOElO};~eri{~1{v ...."-,.h,~,..... u ,~v.t~ geneiiy, t h o u g h n o t m~expeclcd fro ~ experience, w k h ,.~.~.t,,' "~,'" ,,,,~;'>_0;,~~troups, d o raise the ,:u~esliow \\'t~e~c is it all £oiu,::: ~,.~e n d ? D o {he toooi
' " * : ......... ; .....
•~,,~ ~-"~ *i11:tliIK:*I!iSt.-fo{~.i s t a b [ l l t ! , ..~ Of ..... d.~,h~@ C~_l,tiS~;:d b y }h-a;: < } x t S t c l l e e o f a u a p a , . t n e s s relation. A r e t h e r e other intere..;tix~g reI~rtions o n the reals which triviaiize i~ the ch'ssieat case, b u t affect the thec~rv of emma!ire in the i n m i t i o n i s f i c case? T h e t o n e of {hese coa}'siderations Ctl~O;"~tq th'-~t we tea~ to a e o n i e c t u r e of non.-axi~:matizah:iliiv, t~owc,,<~,*' t h e r e ']s a n o t h e r side to tim coin: D E Q is n'?t comp!cte, b u t is com:p!cte enou~?h that i,. ¢:. ~heor~cs i~l..e DF!:O+(t.:ag) which descri{:~" ~<;,t o n t y t h e cardina!ity of t/he ......r,c ..... , a~.z~o ~,~a; cardh~alities of posSible e v e u s i ( m s ()[ the t t l l i ~ , ~ e i s e } ~ t h e . e x t e n s i o n s ~.!;~t r>~: in (the l i m i t e d ) p r a c t i c e (afforded b y Sectio,~ 4) we::e q u k e simple. W h i l e ti',~e~>::.'r?iSbt ~,.., several theorie~ of equa!i!gy of the ~:ea?~< dei:cv*!irw o n the notio~:t i-ff re~).k¢ ,.~,, ~.,,~,,'~ {=t,.~,i,,h+,.>..,be. tiu~t there iia~,e . . . . . . a s mi:iar v. i-.,~-.w , - ~~,.,., , ¢,~ CO*tll~IOI'Isill3lh{),org, q hi.rtk ..... .'; ) ........ : . . . . ,' d o [~. a K d p k e modcq o f equalily h'~ w~i:i;:.i=~eve~.v t w o e i e m e n i s m u s t li~ok a l i k e ? W e l l there are q u e s d v n ~ ot liow tmii~rmi:7 ~hev ~ook alike, f l o w bi-unifort:nly they look alike. Etc, If we insist o n s~tfiCi::nt!y auJri-ut~ifor.m, bomo,zm~aitv~ is t h e n u m l ~ r of parameters left small e n o u g h *o obtain a I,iXe,(:y}y x,vhose, e x t e n s i o n s a:re as limited as t h o s e Of D E e ? : ) : i
7
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2
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';Ill'I! A X I O M ~ , T ! Z ; A t l O N
PR{}BL(E;%'t }?0{'~ ~" ~ ~"~~.
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o
-
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x2.,:c~:[ved 2 0
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b?77
O, h~tr~ucIk~n:
!i~ IS], which we :~,~ad ~efer to as O~\F. we some axiomafi.zatkm,: resuI{s for intuitionistic theories, q"hat i s ~!~'% ,--*,. '"'. ce.rtah"~ theories 'I'~>T> with T~ a:l exte,/~sio'a 0{ To ir~ a larger !a~euave, we dete;mined an a:xl.omati[z,,ti0n T ~ ~f ihe addifiomti cm~se~ue,nees Tz i~ad over T o {n the ~r~:m¢~o ~'~ <: [am~uaee. Put differemtly~ We K~0ved some c(n~serv.aii~>r~, resuits: .~,m~<>~y'i'~ with certain structure is conservative over ~,.~,..jel~,~r,'-0 ~t'~ 'w'i,~a~tL.~L~v,.the ~:~ven' ~t~a'-'m.~co .~>.p......... "~1~2 ex:e~di~,?: a gweu "?[~ hv a good choice ~,aI *" AX{OI11S, tf~ t[~e o r e S e F l t - o a ~ 3 * ~ ; w e d o p:,e~ty m~ch the s:m~e tEim,,~,.........but' difTcre~d,¢,,, Tl~e re,aso~ f!or ttlis new paper is our v,,,,,,.(w;,,4 ~md,,r~*a*~d~"e ~ f ,~,~, o f " , " technk V es used in OAF. iL:ssei~tialh,, ore: method of pr0vin~ vhe co~servatio~a of' T over T[~ amsisted, in showing fiiat the eiemmi~:s ef a representath.% ,?'[a~,~ Of Kripk e models of % w e r e :compatible wkh t h e additloi~a[ Stri.~ctmx, re~:[[th:ed l)',;: T > T h e representative Kr:ipke m o d e l s used were, ,but f0r stJght
0':~......"o-v
those obt:aii~cd from aopivi~ig.llenkir£s complete~.ess proof to Kr]oke :models. As an. a.~'lerth0uglit . i w e .m e n t k m e d. that ~me coukt view Our m e t h o d ,:~.... ~,,,~ app,.~,~,:.~,,~,' .... ~':...... >' ~ 0 f Saturation properties. A f e w m o n t h s after O . A F w a s accepted: f~r m~biica:tio~a, o
•
[1] appeared, 1The ,~':~ccess of Barwise's unabashed Use o:f saturati6e -properacs m the obtaining axi0m,a*izai!t<, resdlts iti classical I0aic tir0wpted: us: to reco~S~-der " " ~x~teo f Satl rat[0~, in 0m" ox~r~ worki As we me~tim~ed i~: O A F , Our proofs co~.~d :~I~ be refo~,mtl~ttq:,| i[~ ierms of sat~tratioa !?'ii>.)~v,t,.~a?": ":') ,_,.,dqFsa.,¢&'"".~:'°i;""' ~ '- ,,~.,,.,...,,,~'°;',"~'-'~"",~v~:o, t"7 moreover, tilt" 'aturatiot~ :prc~_pertieS m:eded are a bit more stai!dard t;Ea~a we su~ec~tedl Ti::[C,.~ ate, howtwer, s;it,.tratidfi properties 6f Kripke n-~odetS wire ~¢,:~ aaat0es for sin tile :CIassiCa.l m0deis'; : : [ I1i S e c t i o n
a~:~d sa:tm-ated inc.,dens. Fi~e exisience proof iS far from novel..but must be mctnded becai,Se o[ the: X'e~u!t~, ~ot Obtai:r~aMe :by appeal tO ."he Classical case~ Seetic,~t,s: 2 a~.d 3 a r e devoted :F,applications to:: the:, axi0matiz~tion pi:0biem; if~:tiiese.two ged{~oris{ 'we rei~tir~c~i)ti[~¢O;t}s to:a
1, w e
t.,,,,~
disct, ~ ,~at'arafiO~t propetfies
,..,.,. , , , ~ , t . ~
0,,:,t~gk:~, ,:.~{ .ih,.~ d , v
..]~,a,,,,~,.t
.~,:,:.~,. . . . .
, e
..,aa ~= .....
adVaidage:thai/~ve*:~,t~ ~i0w refe:~ tLe reade:6to gat,MSe's paper tc (:,,bta,i~:S, betie;,~:' ~}~::~sp~:ctiX'e'for the ::s.i,**!o f r~}uh:: obi:aiijed :and their so:uree~ '!1'~bs6 b f us[ ,~,.h~yac e
!94
C. 5~' ~,'~"~'~;
take,: wi~:h rhx' .... ..creater ex:,~ressi'~.'e ~x;,x~ or. ,,~:_ . tile . {~uii~ionisdc . . . -.u.,-w,,,e~:o...+, . .w.f:~./:. . . find ~he a d v m : t a g e offset b y the reaiizatio~ .h~,, axiomavi:c.:tio::s hi:tee ,-~4~,Sjs ( > i t h : u .~ slmtc,:er, a n a l o g o u s .:~ ~'" "'- p:oo!h) ~n. the cIassicai ......~" ........ .,.;.. ~.a .,~.~ e.~:v .'*~'-"" ~an_m~a~. rich m~ough :,~e? ~2 ~ to begin ~.~i:~:. So. s o m e of t~c mvst.e:v _ . is ~os:. At:caller d i s a d v a n t a g e of ~he p a r t i c u l a r use m a d e of s::turavc ~ m o d e : , is tt~e u o E n e ~ of the a x i o m a t i z a t i o n s , t ~ r ba,ancw, wc otter S e c t i o n 4 wit:: s o m e sin:pie, ~mt :)!e?.smg. results o n e x t e n s i o n s of the t h e o r y of a d~.,.,.aaL,.,~ eqmv:ttvl T h e s e tim?t! I..s:~t .s .fro n o t require, a n y s a t u r a t i o n at am" '~ b u t ~:rise f r o m considerat:o:~s o f Sectio:~s :2 a::d : . S e c t i o n 5 contai~:s s o m e e~.ene,'a~ r e m a r k s ::nd a few o p e a p r o b l e m s . first sppffcatio~ of the tech~:~iquc, we axiov,'mt'i;~e the co, ..~e~mc:~ck:~, - .... -" of :he ~xi.,;temx of a choice function:. Ore' proof is esse::,m~!y *" q 0~at 9,'e gave {~10,,k[?~ ~"
.
.
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:a::ff:,:a:J,e{[':9.1i et{t :/L,tL,.
W e hesit :tie to ca!: the p r e s e n t p a p e r a s e q u e l to O A F . As~:e " ::on: ~" " {he fact " 1!mt we n o w have a vasfly m o r e sopilislicated u n d e r s t a n d i u : of the situatioa, w e are <,.,,.~::c,hesitant of: f!:e simp!e ,.,:rou:ads . that we arc: at tile s a m e "',,ma,. .....wu~:..~' ' : ; -,~ :: sort of i~'~tr~itio:~isticaih:r.Oriet:ted scqt~c.~ ~o .i~.!, N o n e t h e l e s s . 'h wili bc co~:we::e::~ :~ a s s u m e ::~.c readc:" familiar with s o m e of ti~e mate.ria: o n the Cc~mpleteu~ss T h e o r e ' m r e v i e w e d h: Sectior~ I of o u r e a r l i e r effort as well as with o u r sta,udarc~ n o t a t i o n ~.c" 3~:~.,:.~u,m~." 2: Kr}.Ig e"~9 models, tt is atso convet~ieat to a s s u m e tl~e r e a d e r to b e fami!iar e:aough wffh K r i p k e .m~dels so flint, w h e n we assert, s o m e t h i n g u[×~ut the s t r u c t u r e cf the t?~ica~ m o d e l of s u c h a n d such a thco,'v~ we n e e d ~>~t s u p ! q / the trivial xerificatiom Familiarity with I3mwisc's p a p e r is n o t necessary, a u d m a y n e t bc heIpfut for specifics, bu~ is r e c o ' m m e a d c d fo~ tb.: b e t t e r p c r s p c c i i v e offered. Ar~. or~1,:'!©~:ica{ dit:ferc~lce b e t w e e : : the presem: p a p e r a n d tI!0se cited s h o u l d b e ment~.o:'~c,d: ~an,n.. v,..,.,,: ::::'eviousty everythi:~g was cou::table, we n o w find it conve~:ient with ~:~,,~
L
S~tur~m
prot~rti~
o~ Kripkc m~:!~ts
The.re are two w a y s of com~tructim~ s a t u r a t e d Fh-}:~ke ~:o: ~e '.s ........the c h e a p w a y a n d tI~e s t a n d a r d way. T h e chea!~ consucucti0~ consisu~ c4' redt:ci:~g t h e p r o b l e m to
<
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iO ~,4,@.,. ~::Odei:S, ~-~01:t~! c:,£W so:!~td 5:~:a:,:a:}:, t h e ~:',:z:)co~.s-:!:r~c:t:o:is a:°o sot "T'[~8 C i i i ' i i ) :''4~"~'' ,r*'~';<.o~ " . . . ' -p:<,<;::c~,~.'j='= A= ......... ==".....
tho
!~..~',~Q¢; ~:S fO~:iO'WS O~)Serve
b e v i e w e d i s a str~'cIim'e (n .h,:;,,.,,..=<<.>:,vd<-.i...sea-~s~: {£or, as ..:, part:L:.:
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T is ~<,[<.~,~ Q,"""~> ,i-=,=~,class o f 4"<~d,:~L< <';~, ;:~: ,~.£,-.,,.~u,e<="~ ..... " ~t,~,,t-~.,.,.~.:,~,~ c~s~,g~cl.:! tb.eo."y '}7'. @'Or ':. . b.{~ . . . . . . . .to. . . .~ c ~~c'.q:{~S, se<~ c44,." {7. p . . 3 .~.;. ". .,,-]~,-rq.:[>~,. :~'8{;#aC£;~0>." " O~ ~I;~11/ ::#.-.Sv:'[S'01~' K . i : ] k e IY:Qde~ t'hc:ory i o
' "
:""~ .;_-i!t~.~i.tiOl:~]Si:~C ~a S:"i"~'I' ..... ~;>,a:%
,.:,,~,v. tO I ( ' ~ o l I r i t
fo:' t : i o g i
~''+:"
~..,:e ros~iIts ! Y o v © i i i n O A . F .
iit~>
have, however, bee: u::ahie to -o,::,-:tXe cheap co::st:cucdo:} to obta::-: aX +he St~'['vl:2a::10~ o:'o~.81't}e5 We ~,.,~o '" • ~ ::sef::.1. }-:e::ce. w e h:sve ~: c o : i s : t i e r t~:e m o r e cos::~y s:a::dard co:str:~ct:O::o ] \ w o r d o r t w o be1~ore ~'e
cc::str~:ctio:,,:will yield.,addiHo:::.,isaturatior, pt'operties havi:~g nO {obvio::s) ar:a:o,~::
in t h e c:assk:a: case. Let [. b.s :: hmg,aS!:yco A~ :._,-.the,n 0, £s a psir (I: A) o£ sets of se::te::ces of :.... .
/a~
"~S cO';.'Gi.~t~r:t i1 :i0 ~::'~piicll:~o~:,
is de~i,eable in t h e i n m i t i o n i s t i c p r e d i c a t e c a : c t l u s f o r f i n i t e [ " o ~ F', Ac, g ; A . A. cot;sis~e'.:t t h e o r y (.[~, ~t) is com~.:b't,~ if, f o r ewsry sente::ce. @ of L, @ ~ [" o r <# : ,~ T w o t h e o r i e s , (]':,-'A :} arid (.i"~', k2}. a r e CO?rigiS.teg1~ "Wi~]';~g,..,,~,.,,>.,;~@g::,~g~.1~ (.~'! L; . ~s ¢': ~ - ~ / ~ 2", l ~':, A" L} d ~) is c o n s i s t e n t . A t h e o r y ( / - , t k -) s,~ong,!y #x~end:: :.: % .,.:-~ ]1 w~.~ [.,ave boil:: F" :~ t " a n d . ' ' : ~ % tf w e o n l y h a v e t h e firs: : n c I u s : o m ~ ~ .~<~, .wo say ~'}'~a~ .... exre,~as ( [ ' L :Y:'}.. LQ{ C i:')e a s e t o f ( o i l s t a n t s . B y 1.40.), wi: m e a i tl'ie Iar:g::age ' *-" ": " ~'7 t o t.. By an L - n - g y p < w e m e a n a p a i r
r(x>
, :,',,~. :-" (S(.q,
iv;.), 4(x,,.
,,- )I o
,
v . *.a:':::o~e:s o f s e : s o f f o r m u l a e o f L ~ m s s e s s i n e o n l y .......... m,- f-<:,e , " "'~ i~ted arzd such that ,.1 .... : . . . . . q.,). "~(C:( ....... q,)., is a cor~sister~t L({C,I... <'i}5 eor'-,": 2-" .... c;, a r e d i s t i a c t n e w COI]St:{t[ltS. S" iS a. type.C;v*ir au L-:thcorY vI <~', k ' ) if .r(c(," . . o , c,,,.) ,!s c o n s i s t e n t w i t h (./:', 4'), i:~ *-,t~.:, ~ *~ . , . , C,,v~ >' t h e C~'S a g a i n be:::~, d : s d n c t s e w i
[ 96
(Z S;:<'~ ", #~:s;,f
coastm>ts, rix~ . . . . . . xo~) is ~ea~izcd h~ (.F'~ A') by cor~ste. ~*s Ci . . . . . c: ~ L ~i:(17, .~.!,) strongly exte~ds r L Q . . : . , G)~ ...... With these basic defhfitions, we ca,a begL,: te discuss sat~t~ti.e~ properde.- of theories in earnest. Tt~e weakest notion of ¢a{Ui:atkm is that arfsin£ in the:proof of the Completeness T h e o r e m asserting the existence of '~,,imessf~:~g co.:~lsIants: : Defirdtion, L~t (F, zl)~'~) a complete L..(C)-theory. (K A):~s C~ss*~.~>g~-J if eaLcb
~.k2 a mode!.otheorist, C~sau~ratioa is a bit weak to be~tr s~ch aa cxa t~d r~ame. However., too&fie a mi~or exerci,"e to verifv {hat :it coi~~cides with ihe usual defiMtion of C-samrmion for i~Kuitior istic fl~eories (as iu, e.g., OAF), the term is fairly well estaNished ia the !i~erature ar.d we shall ,-e.,ai~ it Ti::is over[: U) c4 intuitionist:[c a~d mode~thee'°~!ic tcru'~{nd[o~\, is r~o~ ~i~,~.'.~x-5to cause m.{:~ch~:
lqrst Flmdamenial Exislenee Theorem. L e t ~,:* = 2% L a h?,ng~.~r~gc,-~j:cardb~,di~y +¢.
L..-theoe),. 77~e~ rhe,,e ~-~s a .hd~y-.C:-sg~.;tm~ged L((?)-,gh,'.°o,:y tl'~-k ~) d:a~ st~v~aggy extends i f ; k). t r o t L We couk~ obtah~ this result cheaply; D',t ff we are mv:m~: the ~tan&~M construc~o~,. we shouid be staadard *t~,.,u~ncu~. ,~,~ { -~~* Hence w:'~~ ra'm~c the class{ca{ p:roof. i o t L-"~ . . and { , ~,,g~},,<~, enumerate. C -:;~:Id tLe set of dis}u[~cto[~s of :~e.~e~,c:~; o ~, L ( C h respectively, a~d Iet '{5,}e~::~ enumerate the set of alI LJ._.>I*ypfewer tl~au ~~ Coustants oi! C. We shai~ ' ~' " ~"* .::1",, by
k:SS thatl -~+~ Assume (.(~,, ~,,) and D,. have bee~ defi.ed f0~........ ~,'~ ,.<:::~-,~ Let
OL0{ e~ ,_~,~z1:27 G{:)~/f ....
piete~i~2ss , .
by
li~t: with:
l~-'~e ;~i~ov~.
A.,:='~:~:,~+;o,- o f
,.-~=....==..,.~ ~(<,-~I '-~ ~{~.=m~o * . ~ . .
Ott%O1
~.=',,,:.,.=
2= ~': ihe
i{-~I ~¢
+,,.{i
?~
=:F: . . . . . . . . . . .
I:l"~J~OV~eI~ "<"-"~a 1< ~L..'tJ..,~c,.i,l ,fii.:,.))~
(' l~ .,, k!~~ } & ,,0) i< al:<~ co,-~sisre~.~L C h o o s e a cOi~.s~s!.:mt st:~ch ~',~i,: at~d caii it Li (~;.;. V:->, [,, (o.~ ~!:XtfIi~d ,L,(o.,~ OV t]ii <'oH
,'~w,,:"~+~e,'-,'A:....
~)0wards
(=r]
--
0;< ~"
>.~31 S~it~:atiOr!. cof~side[' 'G. ~{ "~-~ is a type ove~:°
{"*t~,P.., ~(~,.)I, ~.,t,~ . . ' tiC)i. irt . ~)~..~. a{:/~[. ~IOE. OCCli~.')21~Ifl J?1%r &i3c] L~-?~.... * " [e~ .,"~,,<. be . the .,:-h ...... (:.)l~sKki]i
( F : . i,, } = ,~iC . . ,.,~!..ILI % {<' .,,1. t-,t~z~ '~cv. [("~ r )
the~
sef
r .... ~,.=~
ex~¢:.,>~ ,_ >,.,. D y t h o
o f < . + ~v-',~:{ a l i ,...,~,.;s'-~...- ~'~_b<,ccar~:m~" . . . . ""._ ; u
,;, " r = t.%_ " = ..... :~:'~£ d.>:A.,, : i I ~ -. , <,..,u to-)-
tJ . . (F',
addmo.u
as
Eivid#r~t]y " 1~7~L!~ ,"t '- ~" Y
.... ~ ' '"
i r %. . is. . a . b,. pe.
(~"~ ...... :~'). "I~a t
the crucial t)~;e-:~:eaiiiz~tior~ c}ause -of t h e ~,-,....a.,-o:~5d,.us~ is c o m p l e t e a e s s ..... ,:;,.,~,.~ ,. with its s u b c i a u s e of cous~ster~cy. "M o d u [ o co~?msteac% c o m p l e t e n e s s holds bY the defiMtio~ Of ,&'. Consis~:e,~cv h.h<.ws ':,.~'- - ..... f a ...... ,,,.. the eo~>:<'f.~,'m~ Of
"s ......
---
T h e ~:hst F t m d a m e u t a l Existe, nce Theorem_ ~,.~¢vo~....uS a !a:~'ge. >.o..<* e" cz~ f ~...! y s a t u r a t e d theories, i n classical rnodei .theory, this is [he s a m e as the exisi-er~ce 9f s a t u r a t e d m o d e l s : Iu K r i p k e m o d e l theory., a Iittle m o r e w o r k is required"" - - buS" ~;ot much: A K r i p k e m o d e l can t ~ v i e w e d as a c.ollectiot~ of s a t u r a t e d ~heo!ries; ~:, " C o i e c t k m of f u l v~ :_~i't a ~ : u"~ th(:::O1Jes, s a t u r a t e d K d p k e m o d e l wi![ b e given a s ~:,,. " {h~"~........ Sc:me ~lotatiom L e t K = (K, .a~ , D.. T'). he a K r i p k e s t r u c t a r e for ~~t,~ Ia...~+>~,.~_-~,,..-,,~a L A s s u m e : we h a v e a s s i g n e d set s o f "c0nstgmts C~ n a m i n g the e I e m e n t s of D ~ acM t h a t the a s s i g n m e n t satisfies file o b v i o u s c o m p a t i b i l i t y e0~Mition: {f G ~ ~ and <,, m~mes x<~ I;k:<. r h e a c..{5. (~,. a n d G still ~'mmcs x at ,~.S E a c h a: e: ,.;< detennh-~es a G , - s a m ~ i t e d L((I~ Ldm<.,l"y (!~. &,): K,~<.
. . ~ "" . -. " . ~ ." / " D e f i n i t i o n . A K d p k e m o d e l K is ":..........
,.... sm.ura~ed,' '~ is mlW.,,Q, "
. ,~,~0~.~, AA<, O u e ¢7m a r g u e l!mt a s a i u r a t e d K d p k e mode~iS!~oUkI *oe t e r m e d < ~,, S a t u r a t e d K r i p k e m o d e l to p a m t l e l the futl s a t u r a t i o a of the theoxies at e a c h o f ~Js nc, deSl Such t e n n i a o l o g y i s :Un~'arabiv inflafio~mrv: N o t 0 n l y Wo~.:iid :~hat = ai,m sue:~es{ C a i i n ¢ all Krit~ke = m o d e l s S~it{;fated (:i:0 p m : a l e l :tlle"sat,aration Of the
d[
7
Detlni~o~. T h e IT~crea:vi~ ~r.?;z:~+di,+v_z~(;2~,~&rb:~ oi:~ a .b~r~p:~e m o d e l K is the following c o n d i t i o n : ICC. If a < ..... 8. t h e n t.1~, camd~nalkv of D a ' iS . '""~ ~v.:s. -*ha~:vthae of D~J. E s s e n t i N l y t h e r e m a i n d e r o:f the sta,adard cousmictim~ col:isists in e:'..~p}eying .',t;e F h s t Fimdame~:ttal Fxiste~~ce Theo~'em . . . . . . i~. ii:,:'.. <~,". ~.,~,v'," t c~i!the r',~,~,,~'~',' , . ,~,,1,,<*e,,-' ,~ ,,. ;, ' Filc,.~r~2In w h i l e also wot'~k i~e.,_{!~e I...c,,.{ ...... iv~t;~ 1he cor~strt~ci:io,a, l~hc 'u~;e c:?~'Ib¢ t:q~st: Fm!d t~Ic~>. tat Existence T h e o r e m w[l! g u a r a u t c c salltr:ttioli; tl:l~: prOOf 0t7 the C o l t l p l e t c n e s s T h e o r e m will add a n o i h e r saturati0p~ p r o p e r t y {not nee',~ed in the p r e s e q t i_:a)cr:~, a n d allow ~or t h e satisfaction of fl~e ICC:, ~,~~ ,':he t :C,~ in the ........s-,~ .... o~: sat~e'ation,. wilt . . greatly . . . stren~the~l . the p o ~ e r o~~ saturation, > b r the s:i~e ..... of breGl:y, we s~ate t!~e folk~wi'a#~, .~ T h e o r e m r a t h e r weakiy:
S ~ o n d F u n d a m e ~ ! E.5stence Theorem. tl_I-LH,. L ~ ~F:.A) ~:e a cc,'~s~:<-e~..: L.,U~eory. 77wn (F, A) has ~ san~rared ,~;,:odd sa:~is~ing the ICC. P r o o f . F o r c o m p u t a t i o t m ! d~nmt~:a~...x.~, ......~:" * ~, ~¢ let L b e c o u n t a N e "rod let Lb~;~ -' C~ ~7..o • be sets of cos~s~,~nts such that C,~ has ca:r{~m~.hW -" ~ c~ ~" tic,, '.... A,,~ ... be a fixed ...... m~x,. "° " ~ N,.~.~ ...... ,7 e--I ,~ c.o--satte:ateo s t r o n g extensio,~ of (I2 J)° W e defiae K = (K; ~ D, ~b.'} as fol~<';ws: K is the set of fie.ire seq~ei~.ces,
such that (i) vl'o, ao., ~s the t h e o r y chose** ~bove t.+.nj ( ~ , ~ ) ~.S a fttlly-{~-saturated L(~)--Iheo:ry. T h e s e q u e n c e s in K are given t h e u:mal pm-tiai o r d e r h l g of iinlie s e q u ~ n e e s an
ia (*),
wbe~me (F<,:,& ' ) = (F,,, .&) is fulfv-C,,f =~:i ),:~'>s,amra~:ed; w h e n c e K is a s~mm:~ted m o d e l of ( E A)c2; (Ir)~; Ao).: : =' T h e I C C h o l d s trivially° q + e . d . ?
S'c'~' .~i~g~!~:~,~
? , h ~ "~.,W~ ~-~.~:~'~
~,x4
to~:" ~ ............: ........ ~i:~"
~ K '~-~ b >
i 99 °
. .
\Ve ~o[e ,,.L~ [,.oL,£:.., A~ sat~.~'at~or~ pro!::.*~;~aes of Z~[:
.~),L'g~
:
(
(!~,
"
k:. )
'
'
:
....
{;i~
"J'a), "" %
"-r-~
......
, ........
b y s o m e ccms~:.>::t i~ ©,:>.
K a~id the. '{CC:: 1-:<~r S < a' t~e c:.m1{-,:ai;~.:~ (;( ~="-~ . . . . . .
a~Ao :we
" :~.,.,.Z,:.~:,.t:....... .c>Z d~e %'I.... }q~.al!.,.:~ Pro~0e:rty [i~i ~s a~: :........ a ': ....... co~tseqv~ef~ce of fi~e co~~s¢:r~:cdon Used :i~ t:~...~ comp{e.teness p:'oo~:: K c:~ is as ~,~ ('~;) a b o v e , we ca~: t~-ke
w h e z e ([~,:~>-;~~t~>~:) i s a p y {uIly-g~,.~.~saturated stror:g exte~sio~l o/: :(~" ~i'~I~'r(c}.,
cec,.,+,-'(.,..
P• r o p{..... e r l y.. t i I. is .a geaerali>-~tion oe the ~r~*, .....~Y • u~.e~, ,~.. 4 i~ o:tw p r o o f % . . . .0. . . a. . F (.r[ the . . . }.'-~ e'~-~ 'va,a t ) a t e n - S , a t m a ~ Tb:eorem. It wiI1 not b e used i{~ t!*e prese,~t ~>a~er a~d iaeed ~,"~ be d i s c u s s e d f~rt}':er Prope~:ty I is basic a n d n e e d s ~o f,,:~rd~er e.,@a~atio~:~ . . . . , ~hat a! "cadv' ot~ered }"rapo:t'ty i t is fh~ c~"acii:d saturatio~ prope~'ty !\9r w h i c h we gave the sta:<'~dard consu:uction. Whik~ F r o p e r t y I a n d a c a r d i n a l i t y - r e s t r k : t e d P r o p e r t y iZI .....'~.' a m o m a t i q M / y in the s a t u r a t e d m o d e l s given by the c h e a p col?stt't?ctio~l, P r o p e r t y i~[ d o e s n o L W e wilt use i t in ore" couslrucfi('ms ~:o a!low ~,~ to cc:mpb~te tt,~ c o n s t r u c t i o n at a ,:~iven levei of a m o d e l b e f o r e !?avi~-~eto c o n s i d e r ~ bi~4~e~"!e'veL
2. Const~luen¢~ of the e . ~ t e n c e o f a choice fttllc|ion I a classicai logic, the exis~e~,ce o[ a : k o i c e fm~cfiO~{...f~ VxR*t,.:~
: :
is col~Se..r'~*ative o v e r tile metre e:x:iste~ee assei:'tio~:l~ : Vx ~vR;vvl . . . . . . . .
•
:
With: intuitiol~istic loaic, the s i t u a t k m is m o r e c o m p i i c a t e d : Esse~.:~:da!i){, ~ae .:ea~,o~ is t h e m m d o g i c a l n a t u r e cf i n t u i t k m i s i i c equality, :I~t,aitioaisi:ically,' e q ~ a l i W i s
oftt-r: a d e f i n e d eqt.:h,a:e::c~ f,qa:.ioa:. F,.g:.. ::~': 6}¢ ,"~m:s. ,.icwed :}s co:}s::m:.hv~"
("~'~'1,,' S¢.q:ie>~C:2S~
{t :8 ,
~,. ~-*-..., ': ....... "- ....... :*" v
Of .... . v : "~ ? ' ~":"
~:itloliL NO[ ev¢IW, ftIItCt{O[: O:A eo~'IS[l:[i,'.:t{v<: C a u c h y s e q u e r : c e s p : ~ s e r v e s "h,:~:. .<,:b-~:,~:"v<' ....... rdatioru
There
are
ex:e:~sio::a: a a d
:~o~>e:,ae::sio:lal f::nct~o::s~ If t e e
c::o,~c:e
fu:~c~ioe, f a b o v e ~s ~:o: r e q u i r e d t o b e ex~e~:siOnaL i.e. if o n e d o e s : l o t a } s o a s s : m a e t h e a x i o m o f exte~siona;~itv,
v x y ( x = r - - ' . i X : B')-. tile:: t h e co::se::vatio:~ r e s u : t st:lit ~:olds." Co:.:se:va:io,,: -~~-.L,_,~--,~ : ' , ,,.~... v ~xh~,..:~ , , w e assm:-m ..... ~ ...... is it?ice, eo~:, dec:{<{ bo th¢: ~'ax}om o:e ,~x<¢..k,~,a.t~-.-prov:ded tg:e:e:ma{~ty :¢. "" ~a,: "~%-
b:e: Vxv(x=y'~,'-x=y). y e..',~,. . . . . . . a. "~,~ ~,'
"
o
::ecessa:°vo. a d d i t i c m a i t~_xioms a l e ~.-:~-~..,~.:1""'b y a schcm(~ e x ~ r c s s h ' , a {he cxi:~'~c~ce .of
exten°,.m:e th:ite ex~c::sioaa: choice: fu::ctk>r:s:
:,i ~
. .
.
.
.
.
aathor (OAF~. [ W h i l e
.
.
,--:~ .<.:,>,<.~.:~ aItd a l m o s t
,.
M i : : c h a s am"~o:mced, h:: r e v i e w s ,, "~ [.r,'~ t h a t h e a n d e~e
_ .... ~ s t r e s s th,~t w h i l e w e k n e w a u t h o r h a v e :~,mv:ded "d .c:,s ,a .:-..-,-14?~¢.:~n w e shot. . <'-, ~-{:~:~~ec©.~ks{iv o ~ :},:avL-~.a t h e i,l~:i[Fix sho'~,r~L w e "~ve1~©sio\v t o ~e:'di~=e t i l e s i ~ a i f i c a n c e
of t h e ¢x:e,.:dA:l~ts, .......... : ',~-. p r e f i x - - i . e , t h a t
o::e
an:is{ a l t e - n : a t e
qua,ntifiers
and
m.;,{
m e r e i v a s e V > : : - . . x , , 3.:,':" " "y,..i T h e n e e e s s : : y o f s o m e a x i o m a t i z a t i o : : .........Lc. t h e {IO.* l - - ~,'" OI
~-~ =S<~ . . .X'. .d 'd"<"~
t
of the existence of a choice flmction ove: the mere exis:e::ce . _
assert{o:: ..... w a s {irst d e m o ~ s t ~ a t e d
ia [4]. A s i m p l e r e x a m p l e
w a s o f f e r e d h: [5:].
~: :.,.:,c - * - u a e d as h~s .R. a {'or:nmla it; t w o too:indic predicates and a propositio:xal
col:slat:t: ( }:,:s,::vak~ ~:scd ~:~,h.,":",' ,,: m-~:',a~,~:'the ];,)::i::m~:c . . . of . eqt>n:ity . . . a:,w.mc:itcd by :~-::-'~.,::.~ co:~stanm. ::\::yo:lei'am}[ia:";vit:h:he i::'t:it:e.::istic theory of cq::al{ty k::ows that it ~s . . . . . . :..:~.........o e.u,?ugh wi,q:out a h t h i s a d d i t i o ~ m ! s t r u c t u : e , h : d e e d , w e o f f e r t l l e Counterexample.
L e t E Q d e n o t e t h e p u r e f l : t u i t i o n i s t i c t h e o r y o f equ:alily, T h e : :
,~X.,r + V x ( 7 x : . t : v ) + V x v ( x
= v ....... Lv - ': <'
CO :IS,~:f V::{W/l: over El.? # V ~ ? b ( " " x = y < z
t= .....("
Vx = "~ y ( ' T x = v)t~V..v: 3 v : V x , ~ v .
"
m : ~ =::
{:,:,.-TX>
::
Y2
:
"i
P-~£ ~
.K'l;k) k~t
- = ~. . . .
"
(; ~tg~2%~ ~,G.~ ?~'Z~:)'/S,~ ,t), ~' ~ k ~ £ ¸ 8.~ ~
S [ ~~c(:'-'
:5~).~
. . . . . . . . . . . . . . . . . . .
:iS~=~a]iV ~Wl ~;(]uiv::lliellce r e i s i ~ o n , v~,.,,, D..4~.= ......... ~_,=,.,"~.=~,_=q~d.,=HU:" " : ' I'X_,.,:' K.~ k A e : K~o~Rel !~'~: m u ~ ' £ o f eqkl~vaic~ice rsiaIiot~s ~>~,.o',a t h e c2i}mlHas" " i)~ ~.]it
tl~,%.
=r~
I)&TWiP/
'] t:,-: : m i , , [<-)qk:~.ire~:,erii" " " o ~ II~o~7~" i,s
~}6rsKY~+II;,:
:'(Vi*:~L *; ,~ i.~ d. d. k.i o. ~ l a l t,,,:,~,:.t.,.,,, ~.[.o aim s,'~'wl.*i'--o,'-:q':>:~ e,q~,~iV~(!il#~]Cc~ "~,.,,=~,... ~q':;+:'}"~ . . . . . .<:,}s ~ ' " 1: "b~: = i}iiitsi co{,:erttense.
i.c;. slbstitl_~t~.vilbi . l'e[atlot.s. . . " ' :I' " °l%nt}iio "
8~.: eq',~.raii~y, !..... 'be = i inc)d~i~.,s I t t h e l °;
e. i_~ /)o :. :~ FormaH
7. £
iS c~e:~..-t.d ,."v"
C,
-~. t q,
1[ iS dC;.ii'iV 3 IUO(i8] ° O[ ! , O Vx 9y(-l
3.i?.(J, ::dilCe " ct,t~, -' ;-' -]'4;" = I .' -SD'. = / : , R is a l s o Ik ".-~a,!',el ..... :,:
x == y } , M o r e o v e r ,
:
a%I/'Vx~ Iv~.. Vx:~ Iv:[--'~.~ t ,:= ::,,,", -r,-.,. =. .V>A(X, . . . . .= . X. ; -..*- ,: ,, -- ) ~ : > FIt',
aSsm'ne~ tl~c o p p o s i t e a.t]:d l e i :~.:~= x+. b. i . ~, . I , ,'esPecti,:sly. '~-~"=~,.=:..,=y :
d. c. r e s p e c t i v e l y ; ~:¥;i~
y~ -rm:~si: b~;
but
=: b - - > d "= c,
a
si~lce
=x =,'..-= ~-~, ~ ; l : d
:-'=it,
We shall state our result in some 'Le~ T b e a n i n t u i t i o t i s t -i c with:cq~:atiw
atd
Ieneratty.
. g~a<',e~ L_ tIieorv.. t~.~ "'.r t h e i s u a l s e ~ s e } f O : , : m u i a t e d x~s a a ' : ........
l e t #,'_vl; • *a%.~ ~ b e a f o : r i m t a o f I.. wi[!~ d',_e ~=;,,~. -,,,,,.==.=u ,,,:is .::~. . . . . .
v,~, ~ m~d l e t l ' b e a n ~.]ai~,,: [uriC/iOn] s y m } J o l n o t i~;~ Lo DeVOtee ' P ' 1:o b e t h e i h e o r y :k~ i.. t;i {or} o b t a i n e d
by addin~
t o 't" t h e a X M m S ,
V.~= " ,..l.b,,." ~ ".-;'2 \1~'~ X Vx~, ' :
:'<:,:/4X~ ....
-'~:"~'~-YI' " "Xt ':'=:'X2
x.,. t'.~ .... ~,,:.
tl:~XJ
''121
.
: (t.. t: i
=
:
i
i
V:v F "* v', 9v," ' .V:v }- • -x.,'i, 3 ) ' . , [ ~ ~h.,
=:,
~
.
.
.
.
.
.
i.e. (P" h a com>,r--~i .... ~:vtcv~'s/o.,~¢;;~T+(-'.~ ? T J' exte~Ms T-.}: i ' , ,,: ~ , iS :t l °t Y.laL P r o o f . "I " ~~;0.t ," F o r a , ,a,lc.na: c o n v e M e t t c e , w e s h a l :~ssum{> ~>.a{ p R. L e t ~ b o a sense[Ice' of I., W e sh~;tl s!~ow h o w to co.qvc~1: :~ st~f~ab.ie s a t u r a t e d tree m o d e l of (T. {@) w i t h the ICC' i m o a m o d e t o f (.],,i ~.,.~ --~" 'IY~ Q~is_ e~d. Iet K = ( k \ ~ . IX I" ) be fbe ..... a,~.v~,c~'t o [ iT. {~}} co~istructcd h~ I~e p r o o f of t h e S e c o n d F u n d a m e p t a l g x i s t e , ; c e T h e o r e m . A:s ,aswd. m,, ,.tern tixe o r i < m ot (K, :~-.'). M o r e g e n e r a l y ,
.
¢>
=
T o ~.u~, ~ ........... .# ::md~i~.uitv. ..... , the i,~te, . . . . -f,. r. .e.t.a. t i o p of e q u a i l t y we slm~l use "'%, " to ,-.. l c no ,... at <~,. t hes ~ea.. es - : , " wifi'l ott y two roles: to d e n o t e i d e n t i t y a n d to be the f o r m a l s y m b o l in ]I T h c { e i:~ m u t e thau or~e w a y Io h l e l p r e t f:uncdo~s. ' I b cow,sider a t:mary (say) ~w?ct~on to be a b i n a r y r e l a t i o n S satisfying t h e e:xt,~a a x i o m V_vN! vS~.x:y:is p r o b l e m a t i c : It explains f u n c t i o n s i~ t e r m s of w h a t w e Mready ki,~ow a u d is as e x t e n s i o n a l as S. H o w e v e r , if we are ~oi))g 1o co~.~s~ruct au i~terpret;:~tk.m, it wilt b e ~..,.,,:e,-..~ t<~L)terpre{_ a f l m e t i o n c(msta~t by somelhi~g., l i k e a f u n c t i o n : T h e i n t e r p r e ~.:~:i:)',~ "" ~ is an i:,u:.texed family >e ~ , ae
~
-
,,:~: : Ore ....... .[9,:~'. s~tisfviag . . . . ~ ' c:: ,]..... T o g u a r a ; l t e c t h e a x i o m of e x t e n s i o n a ' , { h , (~ a n d ai! X; v t~ ,L~x.
:
i
}~ ~s ~c.cvessarv m*d suftCient that. for a]:]
{
form:
,
<~.
.
~
N*~ : ~ , . : - ~
~r@ 0 i "
yo
consta~t reatizing-,'~,.t,)'~'" A s ~,,d~,ctio,, hvpotheM~ we t~
where m is mi@mff~ such ~hat % ~ C",,. Sinc~, % has an e:m.;:.~L~'second com:oo~?~mt, it :[otiows that % is prcserved ii~ goh~g f ~ m a~, to c¢:: d~ reNizes % in (.Fk. A.~,.).
I ~v~:t V ' V ~ B v V ~ , ' ¢ ( * , , I°2(V)
{X%Y: Vx ~y,y=~y ~
}, f]~(%)~,
Cr'=O c > 0,
Let (r > 0 'I n.~. mc ~Otonicity of the .F:.,%, O < 0:=~ r;(d~3 ~ I%, (.%,,.
Lj,,~..~r~,£,:t,.~
readily vietds the consistenc~' of w:M~ . . . . " ' f o r a~y o , m s t a n t c ~ L((..,,). Hence. Z , ( y ) = (]~,.(y)~ ,(% is a t y p e o y e z v,. ~..,.,.~,~,,.~- ,-~:~' cas~" w h y ' r e ,~ ~ 0 i s
similar, but easier.
Si~ce ~:,(v) mentions Only constams c, fo: 0 s~:o" and d,; :fo~. .f,-. . o,~" ~f r~'~e~i:iH~r~s~t: m o s t N, c o n s t a n t s a n d the; ful~.oCT,.Lsaturat~o~ o f
~ "~'~ ~' "~:~" rea~tzaL~i~ ....... "~ .......:a[,, .....
'r,.{,'~?) by a cotlstalH d . ¢ ( , . , v L e t d~ bc~, su~2ll <* ell
Hence, for eve~:y q. ~: tA~,~ w e obtain =a d~ ~ 1A%, which ::eaI!zes %~,p cn.e:~: ; : " w:e i:~j:£!~ K ~' '-~ i n d e e d 0~er (!"~, A~, ") for all ;~ ;:~ ,:¢;.[ I_Mn,~ th~s ass~g.r..~me~L i n t o a model of (T f', {q{})b~," definin~= for each c% (I"
L (c¢~- =-'=d , - '
'
[
:
:
(~,
By const:ructiom 'each
axioms :of:'V: : C h o i c e : W e m u s t show
is a n exte~.s:io~ of ~:
wh¢~Ce ",~e E~a:~¢ ii~de:ed
, ....
:
~o ik V XRx~r.
c~ !~"R ( < ,
,Lc,(~
}.e.
,re !P Rc=d~.
B u t t h e f o r m u l a RGv. occ~ws in ,~,.~,,,,i}i~.e '~1 ae~d d, ~eaILre:s "~L "'Fi'm~/~ i"~c.. ~ . ix'.. ct I~..R," ,~ gxtel~sio~lality:
L e t ,,," b e a r l q t x , v y
Q , %, "*
~, . . . .a ".
e~ ;,~
"7 - L ( L -
}
coP, t a ] n s *~-" u~;, s e u ; e n c e .
RGd~ A
Rc,,.d...
A ( G := q , - ~ " ~ ..... 4 ;
w!ae~we.
c¢,~i> % =: <,.---~'-lb., = ,fi4~:~'Rlk,,
~"~" ~',' "~' . . . . .
C.~, . . \ r .e . a l.< o. h a v e
f * ,, L¢, < "......... ~; ~ ' , , ,e,~.w
q,e.
t:]~
.a.~ :.m app~icatio~ oi: T :, - , ,~ co~,Mder the ]ine:~r ord<>r ,.,~ ~ the ex:istm-~ce of :m i m e r v a l sefector~
1..~, ,~~. of
Defi~tlon. A b i n a r y l:uactio~ f (m a n o r d e r e d set is a n :in~er{::a~selector iff f saa,~nes the axioms,.
',~ .~..a. :~g a= f . Y V =-; V
~: > }..... X :> j k y :> v,
in~erva{ scJc-ctor is a e.euerall'¢~ f o r O<:,A < i~
mk!ix:d~:
Opc~uto~
More
i~: fiv}~:tkm.
is a~1 i m e r v a l selector in tim :eals~ F:om the a x i o m a t i c p o i n t ef view, a n u :~¢~,~;~ ,
for
the formula
~',:
x:i ,-',x:~~, x?,L :
i
,
. . . .
,~. . . . . . . .
i
,
'
the. s , , ] t ~ I ~ ~ ~.~ ~ £ ~
o~i-~ <;.
O~i !i;~¢ar o r d e r a~re~axlomatilzed
Q
r,: . . . . -~. ~ :: ~ ~0 ::,: ~ 2 ~ >
kw £he .~.~.~.,~o~4"~-, ...... ,~.~..,~,~ ~.~..°~.
:
~
~~
~.;:~:~ ~ •
]
A :midpoi*~t o p e r ~!or, *~: sadsfiet-. (at ~'........." ~.1, ,-..,.'..';~'. ,-.~.
" . .
,i:vy =,:/yx ~I'#1,..
&~&
$~(¢~-,, . & i - . - -
~:.....
] ~.d \='-~ .,&"2 ~'~ } ~W""2:' 3 , ] " - .f"~*]'"
mat:~",x o f t1:e a x i o m s c h e m e f o r a:~.~ i n t e r v ~i ae,,ector~ " tl~.e coRi~a.*..~ct~0ns, O~ ( x
.....x 7 A x'i; =: x --,-~.... )I
v.~ ~,
.
Li
resp,:?.ctiveD. L e t ~ s p v ~ s e k~r ~,~ m 0 m e ~ t
~o c o n s i d e r c l a s s i c a l t o - i c T h e p;:0e>f o ; Ti,eore~:~
adap[ts quite easflv to exoal~d classical models 0.5 ciassk'al . . . . . . " m o d B t s o f classical T"i The. f a c t that this p r o o f is not n e c e s s a r y - - that tLe wh~}-,:: S c h e m e ( * )," c a n b e r e p l a c e d t~y v X
pectdiari o,
ByRxy
-2. is a n o t h e r ina~ter. T B a t is,. d~2e:to a
othde ,Pea{Is~ .... : . ~v,~ ' ~ d o :,~Ot >ieed o?
o:f c l a s s i c a l equatitv.~, <,stablÂ,,u~-d '~ " "--~..... bY s o m e
t h e cOnq~licatcd: a x i o m s , tTFor a Sin:dim: e x a m p l e , s e e Ba1:a ' ~" 1se s cr~:er.,:","'~'c:&~ ~. ,< ~ [!:D to
Gre£orv's
simpler
axiomatizatiom
."
~'" : "
: ":; ~:
~-° ':-v......... ~;,.,~~
i t v. a l v a t h e o r i e s . ] ~ I f w e r e p l a c e : e q u a l i t y~. bv~ a b i n a r Y . . . r e I a t k m E " . an{,. .'exte~s~0~"~a' . . ..... .............. . hom0rphism
condition,
Vxy(!£xy-:.~ .E?: :ty,~', w c e c t c i a s s i c a i i v ~hc & ' x:~0B . . . .,~atI,.,a : ~ t~oD:, ' :
[
. . . . .
:
w h i c h :is e a s i l y s e e n n o t t o b e e q u i v a l e n , r r e
vx
~,.,,Rxv. :
: '
: ::
:
: ::
: ::
:: :
2:
'
H e n c e t h e r l e ~ s s i t v o f (¢*~).? Or somethiiW., :like :it i~ T b e o r e m :t c a n b e 9:iVem ti%" ~:ia~icat h ) e i e if w e , r e a r e d e~ : n a l i t v : i i k e . m y 0 t h e r N n a r V :}eiatio]4i 5,~)wevdr :{i~.."i:;,~ ,.as.~t{.:a}: va...~.. ~t h a p p e n s
t h d t v.quah,.v ~s m,:~,¢ .I,a.r, ~,,,.~ a,,..Je,.,~r o ..................................
3. C~msequen¢~ ol tl~ existence ¢ff automorph~nt~ Obviously.. t h e t e c h n i q u e ~sed i,n ..,%:''.,,t:,v;~," ~."~...... , n be a p p l i e d to the a×iom~~t:i=:a~io~ oroblem, f o r f u n c t i o n s o t h e r ,,.hart c h o i c e fm~c~.igns. B a r M s c ,~"I~'~n~,~* considers.: m' t h e classical case, t h e c o n s e q u e n c e s : of t h e e x i s t e n c e of a aon-1~iviaf irrvo[u'tion ......... f o r a i a n g u a g e containh~g e q u a l i t y a n d a bi)iarg ~e~u~ton. " . . . . . In fl~e intuiti0nistic case: w e n e e d n o t use so fancy., an ~?u~omorphism ~o ot~tam., a non~trivial set o f ,~x.k.,,.°s"-,,' ......, T h e e q u a l i t y conse{mences o f the e x i s t e n c e o f ~" ....... ...,,a,~a, m.m>~:g.~:~}~1'~{gm <: ~-, per:m~,ta*.~o~? :~r~; axiomaii,~ed by !~:e s c h e m e . ~ x y V x i Yi ~ y i x'¢" ' -V.x..y ~:1.;, x ,~'-~s = y ,:x...~,u{ .... ,'~ ..... ' -- ~" " ~ ' - " > - - x T - < + y{ = . e v ~ W h i l e i~: elm~sical ~oe.ic '~ -~ . . . . . . . . . .
~:,..:-.'-
3xy(-~-:,x = .v}, guarani{eel,s< . . ~he e x i s.k nee of. {wo d:is~inct . eIeme~V.s to > .......... I.,.~ rod,,,_ , ...... , a thfs is not so i:,a Luttfitio~istic 1ogle ...... as the fo[lowim, mode1 shows: K:
¢.~ } i a , ~ , c )
If K w e r e ", m o d e l of t h e s e axioms, ~he~- w e w o u M also h a x e ([ettin¢ ~, ......"- a,~d s~.~m°ess;e~,.: . . . . . . . ~ the v a r i a b l e s witl~ s~~perscript 2), ~. ~;, w,- ~v, w~; :~?:{i.,~ := v,, ,'-,,u - v )
C h o o s e 34, Y':'. for .x,, .va as..umh~g the wd>.es <,L c, r e s p e c t i v e l y . T h e n <,~:>Ik.a ,= a-.~e*o k x~ = *,=~ v, is b
cg.,ikc= ~.;-~
~v::, is b::~,-~, }-~,...... v.,. ?!~>.t
a c o n t r a d i c t i o n . ~5~. ~. ~'he aEcwe s c h e m e is n o t
3xy (°'-~x: =
ntuitk:mi,,~tfic~:~lty d e r i v a b k
3'ore
yk
~do~c generally, s n p p o s e L ~i>; ~::~ ~V~ ~ co.nta[ililla e q u M i t v ,rod '!I" a t h e o r y of
L. Tl~c L...eotlseqttel~tces Of lhe: ex:i:;(e~':K:::~~>f )~ ~Oi~r,t;-ivial homomomhi,~mare axi o m a t i z e d o v e r T by the s c h e m e a s s e ~ . ~ ; non-trivial homomorl~hisms:
..... ,.6~, exis, e n c : c~f ex:tendible 11hire
where ¢~xx:F • . : , d e n o t e s an: a m m i : c f o r m u l a w i t h v a r i a b l e s f r ~
i
~:, X~,.
x..
~{t make it on~o:
H I e c-xisDix~cc o f <'~ k } t o f :~i~to-Gt©~'sLdsms~
:ha d ! S-=='~tCi~ ' i~;~1 : l l ! p p l - U
t~Ic,,"I~II
ii sIIr~(IIUt[I is. c : l i"i e d
a 1>m~e:-~<:# I , "add~ e q c : a i i t ' , , "~:"~<~'>{ w i t h
V f .V. & ..
0"~f
b~,i:i ~'==', : ' ' , : ' F : ' , ,
.....
a~Itol~lO'~i~I}WiG'II .
'P
.
-
w < , : i I,~=~.
- "
i!)s@?:t, i i i ,.. { } i i t
"adti~ [ 8 8 , 0 f ) l ! t O t h e R/k:~-
"
"
~
~t i-J-l
d¢,mcri;-
~21iI
=~=~=_~y
±l~z~!cl=..l,~ ....... ¢" ~"0 1 1
~!Q~;It}~I i
i
,L<,-I'!IUI£1i!'I: ..'"' ............ . . . . . . . ':
L " +d, > . ..... . = t l C""
I f V:v i y ( : : = fy)
(HC)t,42)
V££ ah Vx(S.,.,; :-:-,J:)>O V:,'q+, w , v , .
!i!Y II [i
llI[~.,-iiiO',:!_-~I~Ibg]L ~.~IVi)~'I I !.ii.~)0~1' = iiI
6',,=,=,,+;,=~===v4=uP. . . . . . . ~,,~=~t:~ce . . . . ;,-~,'~..~ =111(~ 8 1 i o" i 7 ! £
~"=l;~t':w¢ 11o'41
i1 I11 W
III " {}I: i' 11) ~1IIG('O i i i
~:O 11!\,' (){:iie~' ~iV(:)[l ,.,,,,,,dL~,4~, ~-~I~
(HO>.,~3~
"
(~;O~,'~4.)
Vxy ~/(y = Ix:.
,,................. ,
',.,,doms ( H O M [)-(HOM4"-.' ~ae,~..O, " ,'-' assert tin~i ~:]}e Amct:~o:[ts " " fot'm a g;:o;:rf'~ of ~c~tO~t~O £ 1 h h ~ r i l s a~ld,_ :-s t h e y . d:o ;.~oc . . . . -vlc,,,= q--' +~.=I,~+-~,x.s" '{q'~.cnc,~'~ o f a s~c;~--ti'i"-.,~a i g ~ [ o m o r o f P
ism, arc'. cortservative over TL The official 8.xiom is the i'ans~tiv;.ty P,,x~.~.~.tLt , : , , : : : L :. W i t h it.. we can prove t h e scheme,, expressibl e .inL~ of .o(.ao~...: no,,o,.sea,..~' ................... it".>.. Sm~tences asserting t h e existence Of back-and-forth eXtendible finite " q - , - '~->, isms taking any given elemm~t x to any other give~ eleme.m: y: Vxy Vxiy:,: 3y~x7" • *Vx,.v~;~vi,:V; .
.
.
each @'X"+ .x~ atomic ~=ith variables from t~e list x . . . . , T + (Horn),
.
.
.
vi)> (~-[om}
~ xi;. ;~?e (:t6t~f.Tt© =. .... to ~?@ . . . . . . ']"~~
P r o o f . As bcfoee, w e ~.v:ll expand an appropriate mode] o i" )t',: ' ~ ""' .1{ e } ) t , ( o n { ~ o f ('F * ~. ° ~ . , {~:})foI a n y semence @. t::it'st, ~.~c Should explain tile: .mteru~ , , a t t c t i O:f ,fu.~tct~oi) "~a '+:'hI"`=',-.= ,+.,..<,.,. ~ s g; I t : p i e modetl When we had a sinide funct.ion ~onst;ant ii~ Che proo7 o~ Tl~ei:n:em 9 I v.,e gavc i t an indexed famik* of fuactio~s as its :inmrPretatiOp.o \Vitb fimcdor> vat:iaNes] We are cons~:derirt~ functions aS bei~ag ':objects : . . s(;,~,/t . . and, h;;ce ": o f .a ~~.ew other obie.¢ts, they titi~ht. O l l k ' cOrlle imo: existence at ~iddes cmite 1~:',4.~,,~,,~,~:i-~,,:.,.,~....
i
2.08
~2 .S.+,'im'~~ ~ i~
~
~
tree~ W h e n ar~ o b i e c t of this i e w sor~ c a m p s i a t o !~.~'~;i~;it :no:Ie r o w e ,a'_~.erprct i~ as a partial i n d e × e d fami!v of ~m~l, io~ls ~~ ) , ~ k~a~c :;~ ~-anges ~.,,vcr all the iiS&:s ;it wlfich the o b j e c t exists (in :pa:r{iclular 3 > a4 a n d , < h e ~ t:%:,e f..m~i>~ safi/~fie~ :Ore compatibilky condkiom
]Let K be a tree mode~ particular, w e ba~e
m,~
W e defiae n e w coIlcctio~s of ttmct~o~,:~ +u, ~sla~'~s ~v E. -=-',#<~'"<' : (e~, ~ a a C;~ "' C z Usip,~= the G S d e ! p a i r i n g ft-:nclio~, we ..';m c?!oes~.' . . . . " a~ ,.~"*m~a,~r:~e;'-'~.~,,~
at]d
{]).{is ~,e<{ L]I1 ©{~N.ITI{~F~!tlIO{! o f OIlY -qTW ,d,t~c:~c.,.~ C(),~tS[3IH.g.
E., = { f%':~"q'.~<-x , --- ~t ~*'
Wii
be
"die
{~mction
e~-, %. >
domaim
say
,
~o
D}a',~,
a<+
~,,.
~a~<~,;t~< o+( fx~ e~g ;}~e, COI18Klfl~:; j ;,~.~: f% ~,~..... ,v a.~; , . f!t<3 l!tle~.r'*l'J'13{ioI~ -
vr>e..n co.. we w i l !IK~uctloII 0{1 ey i'l SDC|1 a wSy as 1o g~Hff[trllCC {A,~, ~a, ,1 ,(~ ~'IK;Ip.~Dev, Oi'~e."{o.-o~].¢ alId OatO O%~ a;Kt ])r~>;e'l+YOs the cqti/:iiTty and ;El o t h e r a t o m i c rekttions i~ {~)th ,.'~;,~x:a~"~u.>o-" Fur!L-:er. we sBaii do fitis ili s u c h a way. t h a : tile: f01lo~i;~g=. ~ co~xfitions u ~ , M f o r a l l a, & c, x t--.,fX~:,, = G, : ~'" (t) ~<'""~ ' °'
........
(~) ,f 77 ,~'~(a} = 5+
:
:
[ N o t e that" we. Dave Mentitv,. at1(t !lOt e q u a t i % ,,%, written hered" C o n d i t k m s 1.a~ w i l l further .. £tx~.!r ~Ite-:~ i:}mt -+'i, -= . . . thus. t r o. t (ttOM:I)Z(HOM4"t . . t 2 ' t ' " = (]~":";'~-t and will b e valid. ( H O M 5 ~ w{;. f ' > H o w fTom c o n d i t i o n 3, : S u p p o s e at n o d e e:~,~,,¢~{,,~,< b¢.e~ defined for' all p<:,:r a n d lhai i:t is o.~xe.i'o-ofio onto, a n d l_~,eSe~:ves all alot;~7.q¢ rN:~lioas i;l iX)Itl d i r e c t i o 0 s , ix.~; :for R :a~0mi c a r i d
a',= • l-Rxo
' 'x,. <-~>a,, if- .Ri%;.(rq)"
:
:
T o define .t':~, conslder'ta,,;b,,).' Case i. a~, = b~. :'INert define i £: (x:)=.v : f o r a l l
"'=:~:" # ~ k , {.... ,A!w. ~ ,L :
x~!.JDas,,
:
:
,
:
r
: :;
:
:
dl
for
:v ~!: r)~,
cx:e::ds
.S[:,.:. T E c
:~':',~',:...... o f C a s e s
:>3
"° :'~
......
"
c o ~ ~'rs cc,::d~dc,',: ' 3 a ~ : d .:::so r:::::kc,~ c'c:zrv ~'~" . :: ..... .............~1:-,-~, ........
P ~ : ° $ V,.., %- .
::f .
.
.
c..... o : 1 < ~": t t.o : :
.
.
. .3°
.
.
f h.v "
.
.
.
.
. o:~.c,,..,:::~:: 4:ii} '..:::d :1, ford.: ~:' :~ ......... 4.i)
~?,:{~):!.qJo:: . . . t } a s 1: {)::c~: :...uocs.s,2 ~'~ "
c(::~::~:p0n O ~:I:
5~:,c:,_ " ~ " " ~ "~
$.L
.... qh,,.~ ),. : . e i
p :~:'; p =. i + 2e}. f o r s o. m e bh:ite ordirm: ;:' ::rid h::ut o r zero " ¢ v e m .ue. . . . d
',,vhcvc ~":*,'; '"
t h 6 * ' 4:'.:.:8Iv;v..~,,...,,.'~d~'~:~*'~e'W t {~,.'t. r'~C,~{:{: l l e
be
":-'*~o f :,d: for:~:u:ae w~{h
,~'S~..~'.v :1., i % , q:/ < g , ,
k.~,::~:,2s ............: -
3:rcfi:::°
~ .3 " .£, 2 .2 V.~, }:v~2 .~v:.: :" "g.G,v,. ~x i.d{,.
a n d matrix. /~
r
a ,¢
:
{
">
W:. G,,~.
.....
;e.,}
w h e r e p : , . : . . ~1,. < O a i : d e a c h @ is atomic. T h e axioms ( H 0 m ) a n d the : -,.:,'C:;..... ~::J :~:,,, :)...~ ...... ~, assump~ion tba~ each ,:;v reahzes r ! x v ) o : r cG. rea:~t~:cs '~.,,:x,~ t::. I©0 de.Eh:;c:; acct::.rdi{tg ;-~s ~,' <: p was eVef~ Or o d d implies that %tY:~ " ',* ,s ~- a type. ~ s{ ~, ~e q-~¢ fii:s¢ ctm_S:~:~m* n o ! j~l , { h e l.,ot i ~ te~%;,:,, ¢ -: ~ " ". . . . . sh-:~ce t h a : rcalizeS ~';:,o iSuch a c o :: s t a : t ex:s:s y~c~c~s lhe t x : s t c ~ y e <.~ ,X,.... o~..~:.d~a Cod>{aota :x~..al........ : g ~ ';.) &d;casc :4/i" p is odd.: ;..e~ G b e t h e first c 0 n s t : m t :~oi:i s :the Iist !¢c..,,,;,..<;,:-ar, d •
,
,
.
clcfine: 2 ) • 'r,~:(X:} ='LI-
~2' ; -'" . , : ; ~ ;~, 0~,
:
:
: :
!"~b~'): consistin,a Of all f o m m D e as in ~. b u t widi d:e m.atr[ces, :
:
: ::
21C
C. 5'~<~r}':;xk;
p > , . . ~% . ~, .<.. p. ,~ e::tb~ . . .r ;. iS . .a .t3~e .
list
p-d.,,
we c~a~ ~'ealize it i_~va co,~~stm~t J , ~ot i~ the
.td~,.~,<~,.
T h e constructions, o f C a s e 4 is comp*ee*~d by ~,~{q~,=om-, #:~ ¢,4 "~= .... for air # < S~,_~.~,o
] ~ e Iiew m o d e l , say K H°'`~ is o b t a h l e d f~'em K by addir~'~, d'~e ~e~w d o m a i ~ s 1 D K ~ ~ = F,~ a n d i n t e r p r e f i > g f ~ b y the fam:!v }:~÷> ........
: T h e b a c k - a n d - f o r t h c o n s m a c t i o n of C a s e 4 mm:~ar~tees .t~mt iis hmcr.:i0nS ~[[ m a p Dc~,,. o n e - t o - ( m e o n t o itself. Cases t - 3 p r e s e r v e d:is p w p e r t y . M o r e o v e r , ~:he ....... ~?~.~.>~" r% r- g u a r a n t e e t h a t
for all a t o m i c R, aIt ,a~. . . . . o:oiF V f g x t . . . . . . . .
~;;,,~~ CIr, a ~ d a[~ :~4~propriaie f:'. T h u s . ""
.
~- . . . .
a e d ( H O M i ) is valN~ ( H E M 2 ) foil:~ws f r o m d~e fact tlmt e a c h .{~, is o a i o . v . > O M a ) - v H O M 4 ) tohow h°om the tream~c~~t of' C a s e s I-3~ Fir,,a!b< :{H-]M~% fottows p r i m a r i l y f r o m Case 4. L e t ~ i:~e mh~i:mal sucia ~]~a~ a, 5 ~ C ; . . ~,,v g,',
b ' fali into a case o t h e r thm~ 4, t~*se condifiot~s I - 2 .,:~d,~i n d u c t i o m This complete:s {h~ proof° qoexl.
T ~ e examp,c', ~'~ of tile ~',ve~lI]h.{l~," ~; el'e {hi~-; sectio~t si~owg t h a t F ( ]~<'':>~'~ is ~tot co~servative o v e r E Q (the t h e e O of equa{ky)o h!de:ed~ Li)ch#z i13] ( a n d ~ater the a u t h o r - - - c ~ . ~,o~ c o n v l n c b g l y demonst:rate~S the :~or~>-homoge~mib' of e(pm~ii:y by
V y ( x = y v '7:v = y). It m i g h t have o c c m a e d ~o the r e a d e r t h a t we could have prove:~ T h e o r e m [ m o t e easily by t r e a t i n g all ,of o u r function,s as izr~ C a s e 4 at~d thews s i m p l y c i o s ~ ( : u n d e r comoositio:t~ arid ~.v 7. ~e--" _,._,:qon, I n d e e d , we c:otfld h a v e dm~e this, O m ° 'prooL in aa~isiyi~ag conditio~as i - 3 , wa~ ar:: a t t e m p t to m a k e {&e growp of autom
SY :'.
(UNtC)
~i'~'~. C o n s i d e r how w~c w:>,,~h:i a;t~.m::~p~ to d o this. O b v i o ~ s l y , ~ e w o u l d w;m~ t:o sati-4y a ~ew c o ~ d i l % n : Fo~ a:!{ ~,,, }< c. ~;~ s: < / > a 4, ~'*"~'~(c~= d ..... P"~'>{a:} = e':, <'~,.~ Actuallv, this is m)t e t , o ~ h : : W e woukt ~.......
....
H o w could we d o this? W e d o ~ot even h a v e m conside:r w h a t n e w ii,~t,S~.:to a d d to spot the difficulty: t~i o u r t r e a t m e m al×~ve of C a s e s I.~.4 handI{~,g coad{!~ons 1.,~!~
21!
"~ " ' r " s ; o f h m c ¢ i o ~ s , i t ..... t~**?:80~t lJ~a{~ _ .... s e ( U N i C ) o e o a r Lm,,m,.:,. , " : . IO imp,o, FItUS{ ]i[{IVe e i ' c ~ a t e t :
be,;~lD?~tV
.... ,,. ~* ~
"
~.r*mM }. ~l['.x ~¢~'
h,mctioi:s~ w e
i~l {.~le l~'Oe:'~s'I q ' ~ # d p~(.,-.~,;,,{~a *.;:Gt,,
L,,,.a .......... , . . , . l e t ~s .... b.............. " .... re~- ":~:,o<'o:F 'r" ~-,, t h e a<~,L.w.,,oze ~.a <.~.2{x¢Oa~i?.
oi>eral k)tl, I r,-, :aotc t h a t a ce.~,'.se~'aue.uce o f ( U N : [ C ) is
t:;'x) ! =-g " - i z ) , and exte~siomdity
i~ a l l t h r e e v a r i a b l e s ,
OLV' g r o u p o p e r ~ ; t i o a is o b t a i n e d b y c h o o s i n g a n e l e m e n t e t o b e t h e k ~ e n t i t y a n d co-~.~m~erm~ : A"~,-<" ,, ~-, Oti Lt}@ i e f t by... iV: . . . . e a c h f m v c t i o n .ia:~.~ t o b c- .~l~L_uel~cah,,~ X " y =,
~<¢ x} "~,'~ ._ g(*, xM' **(e -r"-.--"~ x',
~,~J .-..j . . . . . . ~{ . . . . . . ~v;JAa
'~Ve, h a v e :
x , ( y , z)--,-~,~ " - . y;" -~.V-~ ~, is j u s t the; e x t e n s i o n a t i t y e-
,*: = f<:"'"~Cx:) = x .
x"
e =/u:'=>(e)
(04) o f F. F o r (G;:,,, o b s e r v e
m's.: t h a t
Also = ,x.
F o r i(~:~,, n o t e t h a L s i n c e f!<~::' i s a,~ a u t o m o _ ~ p h i s m , '4~ere is s;om e v s~,?ch !.ha* t m " ) ( v ) ,=: e. T i r o s X . y =~ e. B i , t t!~is m e a n s of"(t:,'.)g(e.Y)¢i~ ~ g ~ ¢ @ - e ) f p ' t
(UNIC)
wheace
f '=(t'"
yields )
and
y.x=e,
e#.~.
F i n a l l y , w e v e r i f y ivy4). N o t i c e
2
,
?
{.*~,~.~.L4. . . .
a : { e r , F ~ ' k v :!.." . ~ ,
~'"
~¢e)=:~.e-ay =-]
........ U
...... i c'~,
whence t .... ~ x D = gv ..1, e {,<.~ .'.
fh'~.a.s. ( G i)--(C14) a r e ckr;T~:abk> ..... L,~ "~"~>~l ~. . .kkNtv" . ""/ f o r t h e ~:<.,,,:~,.~~:~a ,=m~{dp[icatkm~
Note, '" +~ :i~.rmer " ~* e~,.o~,'.,a,~,vthe .e~' . .oU{} . . oDe-.':a)~i,o~a is ...e-*" ~,~a~h=,%w ~,,.#~-,'..:, ~.,.o.,,~..
1~©r, x . x~ ~: .e k. ~ a~,,d ;v.,-, is alt k~H~O~!k?i-g;~'ilSP-1, t_et L "t.,c" e. ?',a..~e~l.t~,.~3,."--~ " , ~ " as *~-+:"~..v~c .~, (};d {et :, be ;~ n e w c o , . i s [ a m . , a ~lgw. Dirmrv. ~?i:~erar.io~. s-vmbeio -::-,.~id~:c ,_,,,,~..,..-w ~x-~-,mly fvme'ior~ cc :~.sta~t. % ' e ~-~...,+,~.~ " defi',~.e 'theories" °i{ • "c>~ a',~d T ~: i~ ~.he ? a ~ a ~ a g e s ~~ , .J. . .ie. °} av, d L U {F}, . *-"~ ..¢..>) ........ . . . . .:.' .".' : "~. 'I *:~ exte*ads T b y the axioms (GII-(G4)a~d ({.l~,:v] ~::,s>,;= . . . . . . ~ m g th;.~i . {s a !eft h w a r i ~ m t o,-om~ .
e ~m
~,,~ ~
.
.
.
.
.
.
.
I
t h e s e var{a~qc.s. T h e a x i o m s k~ ..e
T I
" ~'v,. N,IC.~
wifb,o m
YtmcIx),i
ll'C:
(Fi)
V:vy:v ~Tiw(F.vy~.... Vx)'e wilF~e:.~' =
g)
.~'. Fx3'a'}]
(F-a)
V x y - T w ( F : v y a = w-..~ Fyx~v = 2 )
~t::S)
Vxy(Fxyx
=
F(y,
(th;) (FT)
y)
g x ~ y ~ x e y : : : z w ( F x ~ y ~ : = g r x y : : ....... F,,',h )'t~.v = Fx:: ),:Oe).
~"~"~ "Av"~. t~:-'°- >:.:~.~>"~ ~~ . . . . . . a:: ~:,~.q o.'°.'" f o r m a grcnm, o f a u t o m o r p h i s m s : 0 I % ) a s s e r t s f h c t.rals}tivitv .~f this ~~"~ a-, a.~{ ::i: ,.~:..~ ; , r - , . . ~ is tt~e t m i c i t v a x i o m . .Tcv~" is i ~ t e r p r e t a b t e i..'.~ '7:: A{(-.~~ 'we: vtdd a conslai~t e b,, t h e dcfit~i~.io,,~ :go ~-:: . . . . . . ..< ,. . . . .= F~e~:y. C c m v e r s c i y . T v ~s i,te,'p.~ :~ A:Ac ~~ t .... by: d e ! r u i n g b x y ; := v~" "~.a ,.W no{i!}g that ttle: u~TiGue,~ess ~?~o o f fo~ t h e i~., ~:>~o :,~,:{p,:,( ftio tis!ic e l m file irwcr'se fmi~;~k)~, t h e s a m e L-oco~iseque~ces°
: :
:
R ~ w a r d s t a t i n g an a x i o m s c h e n i e o f u n i f o r m homo~:~'~eiw, w e L ~ r o d u c e s o m e n o t a t i o n f o r s e q a e ~ i c e s of Va,'iabtes. F o ~ Z e i t h e r x or v. we.. icl z. ~ ; z{: :det(ote "
~"
~~
" ~ " a~d
z~:~, . . . .
ai~,~ r m p e c t ~ v e i y .
The
sehe~iierl ( U H c , m)
2~3 v,Tmsl;sls e{ ~dl s e ~ t C t x > ;
%¢ .-,£,A , 2 ~
\ v i l h ,- ',~';%:,,, :i**:~
.....
2-,~,
_ ~.
., ~
.,2
:{
"~
.,2
<-e~:-~,./[y~.,)..... ~> o
. , -'" y~.. ~ ':c~,: ::: . . . .
, "i:,,,,.j i ~ ....
(h)
"~, ' :": "~ '.,~.~'< ....... - v., ~:~"= x~,., *
~?,, {?* ~'8.~a£{1"V.{ OVQF ~ ~ "q
:= ":'?P~L
(:.,~.,
}~',A
~ . [ C q ::=:.v~ A ~. . . . xi '~ 5'e~ =: )
,,)
.,,.
,
,
->/H\UvL =: e(." a. v¢ =
.
. . . .
""
,,' • ,,~'"~
.,7.s
~he ...... ~ " ~ e~'m , ..... gU) . {s . t h. e .uniIb~7I~itv.. . .eorm,p.~: . . . ................ .'~~ t - £ {He app:ro:~qmab.on" ° t o <,~]..,~.),/-r j ->~ ",'-~ L<:." ( F 7 X ~md " . . . . . . . ,. T ~ w o , ." d e i ~ o t e r + ( U H . o m ) . co:mpositR>n acc,?~Rm~., ~'"~. t o (,_m~x , r , . , . -vv. g , c ~~<.t {:~ g . e a c r a L m*'~'~<~':"* is a ~ r o p e v e x t e t ~ s i o a o~ 'I "~-:~c'~,~ w i t h r e s g e c ; t o t h e ~ c o u s e q - u e a c , : % T h } s wit} f o l l o w r a t h e ~ ........ <.
H i n l to t h e proof.: E x p a n d
a s u h a N . e ~,< .......,_,<~* o*{ ("i'u>~°'% {q=,}) to e.n,:: c4 (°F ~ ~{.q.'L Defir}e .F b y ,:he us~m[ i n d u c t i 0 ~ . T h e ,':. , a c k' ~- a t d ~ , o...... , . , .p r e ~ x a~iow7 (}73) t o b e s a d s i i e d ; (h~ .... a c c o u m : s f o r ( F 2 ) . (F6'i :,,. (u) a c c 0 t m t s g-,o~,-(F:I), *iF-:':}, a n d (c~. accoul~ts {'or ( F 4 ) . (F5} is r e d u n d a n t . Ge.d. Reversio~a the multiplication ~ th
right t r a n s l a t i o n
g i v e s a corrcspe,,:~dis~ r e s ~ i t fc;,: T ~sR ~b*m,, a ~ . i
invariance
in: p l a t ©
~.~f } e f t
~;f~.~sIati0n
invar;i~:~nce. ~
L,
c e n t a i n ~ ; o n J v e q u a f i t v J w e h a v e b o t h ri.gi~t arid l e f t trat~slatfor~ i n v a ~ i a n c e ..... b v t h e e x ~ e n s i o n a ! i t y o f t h e g r o u p o p e r a t i o n , l~n ~..,eneraL d u a l hw~Lrianee ca~m.ot <~, ~:~;~,,~:~,, be. imF~)sed. ] b i m p o s e d n a i im, a r i a ~ c e whe~~ L c o m a i n s a~. a p a r ~ a e ~ s r e i a d o n as w N I ae; equa{ilv~ R~p a c e ~;qu.:3iitv b y m:~aet,~es:s {e (F*'~ ( P 7 5 > 4 ~, ~ m d (~:~ < S i s x~mald :~t~t~. b e n e c e s s a r y if ~he grcn?R o p e r a ~ c r ~ w e r e abe{ia~% :S~,r,,,,ev~ ~ ,,,,-e .........~.
C . ~ n ~ e ~ e ~ a m p l e ; i'l"~e h~tr~itkmis{ic t h e o r y A G o f abe{~a~ g r o u p s iS r~o~ coa.sei:~ at:ire o v e r t h e i m t uhi ~ ieo:noi s tri c y : ~. G o f g r o u p s w i t h :respec, t ,:o ,,~-:-'..... .~>;.,...~ % , ~ w~.. >. a x i o m a t i z e A G b y m e a n s 0 f_ t#G"[~. - 7 t"£~~ 4'=" )alld : : i i: :
X':y~:y''~
. :.
'
{
:
(A)
214
(z s m o r ~ a @ i
;
If we consider the functions f~z =: x" z a~d/;.~ ::: v ~ z . then x,e get £ y = k,x,
f ~ e =: x,
L e ~= v,
=
.
whence ";~,'~,,-,': ,~; t h e :fir~e poims at~31I{ b~ck--:md~orth cxwndibD e~efixes'} we~ cai~ derive every sentence ~ith .a pre.f}*,
and ma :fix eo,asisfine, or" coa:hmc:~kms, of the tt.%~',~m~.L
•
yi
.....
aI~G~
where e agai,.~ ranges over {t, 2}. In the frace of (Horn), the cxis~emiat q~antifier. ~e, can be replaced by a m~ive.~at one---~ wh:cncc, i~ G, we c:m assume t[~e variaNe to be realized by fl~e grxmp idm~ti b' e. ~INe ciassical rlon-abelia:~ di{~ed:ra~ group D:~ a~d two of its quotients afford us a simple model of G in whici~ one of these theorems of A G fails to be forced: K: ..... f
Suppose the sentences i~1 question were: tree ia K fo,, a, v ;~s:;un,i;lg the values a ,es~:~ecfivelw We consider several choices of .v~ a,~d fi~e co,fresh,old{m- va{ues o i ~,!,
.
=e-..
a d . ,~
-- a...
,'~z,, u o- a:::> ~ ~7: 9:~,
a& a b : ~ } .
F r o m these catemat,ons we ~mn conclude that u is ab: U s i n e tt,ffs value of **, we shall make a few ~urther calculations aBd hi{: u~xm an in¢onsi?.~<':>cv~ X) :=: dO 2
_
~
u - - > a ~ t } h = u : S y ~ is b.
=
. . . . £
}1
:
:
:
:
: ;fOg' l.~t/~ 9 7 7 t g ~t~%
,:v ~.t:.v!{
.
.
.
.
.
.
,!,~g,a Y~!,
....
<::,>ii-:,:~ :-~ x:{<-{,V~ . . . . :: ~(gg) !;°' f ~ O
ca:
, ~-
: : : d}~
m s / i x ] i c t i ~ e Vhe fact thai v,0 ~ ¢~.~,, \Ve ~~..... ,-,~o-s........ ~-,' ~ ~ s u i t a b l e t:heorv T A° <.x~v.~ c~_**Iga ~wen theo>.~ T, ~:.Ve ,a..,~c t~aI \-<"k' b e c o m e s
kQ~xy(F~:~:xy-'= 7he,x)
:
i' >(%7 .,-Q
w..,~ . . . . nC,R._hs .F, 0 umvers~} .... eP; and r i g h t ma~tip~k:a/ior~, .~.::~.,,.,..,.,,;,"~'~ .......~;";'~* a~'td con.'qder a s c h e m e of b i - t m i f o r m h o m o g e n e i t y .
4. ~ m e ~nmsequence~ t o r a decidable e.q--l.~IRy A s ~tm~sed b y BarwTse, l~e a x i o m Schemat-a o n e gives in results K
aild
exteilSlonhllllVthat the: trtlth rabies (i.~, O:f t h e Statements a 1- % a Ikt ~::>)do ~(H
(lii'Ae:Se~ o~1~: ~,an~s ol:,lv o~!e CO}lSta[it' p e r O b j e c t ) ; : t h e Stn:,.Ctm. q ,e.w~,,~):; o r ti':~e forcing r e l a t i o n . In. p a r t i c u l a r ; if o n e starts wit2; a m o d e l : K b~i~e or~ a tree, :ihe :~:i:>::Y~<~,,Koeedure will vie,ld a n o r m a l i n o d e l K N: buil:t ,~m the s a m e tree. { i
:
~!16
C: S~:~'~.~ ~f
L e t (D'..', deno>~: {~.....
~
......; - • ":~'
" g%'¥~
V x y @ : y v - ~ x := v?.-. .As m e n t i o n e d
~,.~a
m. Sect~e,.~ 2, {f T is ~ 0:.eo=:v ¢o~i~amm¢ a b ~ ' ~ r ~
re=:<~on ~y}l£?Oi
R a n d equali%~, a n d '~,; ~ I-e is t h e ex~e;.'_s~o.s . . o f . "P .h y .: ~ e a d d : i t i o a " r -, eonsim'~t ,f a~~d t h e c'Loice a:.~d f u n c t i o n a £ t v ax~.,,.:~ -:........~ ~,.e.,~,..; 'tLe~a: ] ~ ,D~.o:,TI
"
~s co,~serv,:~!~ve o,,e~" T + w x
.l'.,.-
f~Inclion ........V x ! v / ' & ~. ""~,~eo~c~x,~ .........
~
.'~yR'xyl
C O ~.gQCi~.tOI~CC{Q 0 f {'i*e ()xiS{:c:~lc:c
:-:uti¢:0s,. . .
',f
9. . . .~. . . .is <,:'"~,,~,=.,, !>3' ',,.'<'~rk(~< o n e ' s ,x"~x. u p ;.'~ a o r m a i
i a d u c t b : e i y A~ ~ : .....
.
eXiCilS{O~;'~; *_;:hO{Cg
0,!~
A. si*,:~plc- tro~:ff ''~' .,, ~,a,>, ;" indep~;~&x!~ o f 11_m~ (?..f
~" -~a.< K /2
w i t h .x,',-" ~-. I O i'--',l~g~)£C:{"
ire{; m o d e l
K aad
" ;~ {"IIi!C17i(HI ~, -'O~l~Zg:,~"~ j[
T. h e r e a d e. r ',~.",-'~vec~£rries . ()'-e~ . s u c h a constl'ti(t{ot] wili n o , i c e {i..~[~[ ¢~'e'r:" . . . . . . {s !~Odilk{~.¢'*" problematic
abort!
~{ at ;tli ...... ~hs ~, <)(:it~:Ii~iv ;~.s [(~c~;~{t\ i4¢c~t,' . .t:.c, . . .~]1~v . . c}m~c,~/:r':", ,,
ca~ase:d b y t h e ~4eeess{Vv o f scttisfx.'in,_l exter~s~o':~a~i-b:. A n o.',.?ier . . . . . .~ec~>=_~,.,.. . ,'~e~- ~*. ~rpp{ies . to. s h o w. ;:he {loI'~lo~e~ie{tv~ . . r%.,. & ~ e c t d a b i e e q u s ! ~ v . Le~ 1DEQ~ t i l e Ib.soCv o f a ¢,~=,..~.~,a.,ve .-~...: a,.~.~ :. eq,~a~ity, ex~e~:d {he ~b.eorv . E Q . or" equa.Ht3 b}' ¢~e ,,.=...,<.~'~¢~,=;*' ~ ;oi2 a x i o m (DI-. g ..... ~, DEQ
........ is CO~I~SI!'%~iltl~ :() O)'ie?i" D E e
T<" :"='*'='~eVhiS= v~:m ~iv acv~'~: IL~at i :q~ ( v u t o . m o m h l s n ! o f n ,]ocr:13}, m o d e } K o f ~~[-:O iS ;'~+ =~.~,,, a n i n i e x e d
! a m i l v 1~: I
- e l" pe'a~.ut.qtk'ms o{" ~he i~dividu:.:!.I u@m"% ._..~,..~s O a
i n d ii. i~,' (sat{sii=~nv ~nc "' ns~.~:I{ i>.c ' ". .{~Sl~io>!. ... "
!~L
~iI} = (&
]~:: r~.~ ~, i h c (,~' (-',.-'ilS.I3( )S I' i"i O
1i
fs/n{}y
x:::::b
[
6:v
~£
.v={e
is k~sI s u c h a 4. o, :'~ s so.g'i o{ a~'L*?31~glv:l't{ "~" et' .,. ~. ( ( .~S' FiO[ WOi'k 4,.,,,~ cOlisi('~ci'il]'> i~ L"['O~,IDo~)el'atio~ .,~,) . . . . . .,~ . . .::,,v,;,:,~:, .... G {' (|i)I =: t)F:;,Q ':~:~"a n d A:G + ( D ) ~: D E O A 0 ~:
-~ e. . . . ~{ w~=: dclmc'~' t!*e
s c n t m ~ c e s (.e > 2),
E } , : ~ x c " ":r...(~:,~. x:~ e x } t ,
......
,
t h e m iu D E e e v e r y s ~ n t e n c e is e q u i v a t e n ~ to a p r o ~ x ) s i t i o n il ~x'i,--/~q i a t i o n o f E "s o. r. .o.u I:)s a v e ¢ 0 n c e m e d , t h e O b v i o u s . c a r d i n a l i t V# . r~.su ....... l. "is L a e.C x,~here/, ~. . . . a. n ~ e : s
£heo~em:
[fG
i:s.a s u b g ; o u ~
~... ,~:~fi~t,:',.,,. :o.,-q,~.~...r,.,,.-}£ th.~-~n t b c orde:r' o f C- d~vid,:~ '~hat ....
o[ **
W e def}-v,e ax[orus.
e
"
"
..-Fkk 44~-Z).
LqC~£~S~O~'H
From
.
.
.
.
m
t
~,
~
-[
\
•
.q
.
l.,i
~"
"
U,*4-~ i t e a s u v i'o[Io> s t h a t (.~ + kZO r (~.mg). ~ o r . assuage iiC t o b e a h , o a ~ o.0 {.;
~. ~:' ..... >~ ~. , . .~:., h
'~ L > ~
...... > He~ce
we
have
~}[1~.e~ f o r 8O~A:~,6
~e,.°*'o'"~.,...~-s (&,,,
'I'hco~'em, H e n c e we have s h o v n G + t.E ~ to e.xtenc~ D E Q + {Lag), Nc, ,, ~.om].~ , a t mode! K ot" D E Q satisfies (Lag) if~ K sa:,sfi~.. , ~",'xt, note that ,:~. ~ *: .... ~: v~ f3 m~4 D m D~:~ iiuite =;; card i.b,7.., i carc~ tDFb'
( '::
Half ~f this we ah'eadv, know: Otw p r o o f that G +-(D) k (Lag)~ c,epe~eea . . . . only on the satisfaction of ( * ). F o r tiie com,~erse., suppose K is:a normal m o d e l of v.~ (* a~¢~,. a n d that ¢~< ~ wifl:~ Oe~, D~tl h a v i n g cardinatities ,'~ < m. T h e n
yiekl a%/~:,,~-.-~ ~.,.o.,,.,,> B u t ~ Since the cmdi.~m~ity o f D>:> ~:s m,~;t~(m.,n)~, we have m = ~(tn, n), i.e. ,,! ..... i.e. ( * ) h o l d s , a0 ' t ~ complete the :proof, let K t m the saturated tre~.: m o d e i o~ the v>_m~ of T h e o r e m t.l~ Wtien we .norma!bie, the cardh,Nities ~' d.onm kv~s ca~, ~--'~oc~,.~.:~,.,,;., .... b a t not that nltlch: T h e folhywillg holds: "
-%-~e
e < ¢.~ and D a h a s iW]nite cardinal ,~: ..,~Dp :[
[Note:we
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:
c o u l d as e a s t' °D c ; ~<*~d . . . . . . : . . . . .]'i}.~;X)~Itl .. ~°*
~ . . - L ~.~
La~, v o , , c h m , 4 t a , :
:
. . . .
a,~t., "1 , :'.t: & ~ l' i d' "gi~u . . . . . . . .~,b'[I~,,,a~th.~Sa ...... <..., .............. A
,J,.,,.h~,.a.=,5,~ ......... },~.a~,
mode.ls to satisfy: (* * ) , ] It iS flOW a trh@d mattc:r t0 define abelian ~7 £r("V} opexatio~S on t l l e d o m a i n s D a inductively uP the t r e e : G i v e D~.-'.:o a n y abelia~~ g r o u p structure a n d proceed f r o m t h e r e - - a n y : abelian: ~ a o u p 0s~ L,+'Cg ~" C~t,t " ~ 'k)c, .... extended t o an abellai:~ g'ro~ip. 0~:~ L)i3 ~ :-:: ":' ' :~ *" "" *~"" ; " ...... ,..,..:.Ax
2t8
G Sme;ynski
. '-~-~ . orde~-~::d . system. ..... Of .~:rol]es. is condition ( * ) or ,( * * ) holds~ " , [~c - ~esu~t:t~.~ .... •. !:vart~d~v eas~ty seen to tm a Kripke model of A G + ( D ) . q.e°d. 1 ~ e ~ r e m 2. D E Q + (]~tg) is decidab~co Ratl',.er than prove T h e o r e m 2: we refer the r e a d m ,a m~ ~ pgu::,f of the decidability. of D E Q m ' ~e] ~-i fo:r a~ example of t|~e teclmiqtic ~o ~ nsed ........~t'fich is to reduce the decision ppc~lflem for D E Q + (Lag} via Lifschitz' quaafiiier etimina-. tion for D E Q to that of a prog~sitionN caicuh~s whose &>cidabitlty is easiJy estaNished. O b v l o ~ i y , similar resN~:s a~e obtains.able for oilier algebr~qc :~bjects~
Exan,~ !. "1-.he theoLv of s e m i g t o u ~ with decidable equaiiiy is co~servat~ve over DEQo 2. The ~&e:odes of rings, co~mnutative rmgs, eommuta:ive ring,s with utlJt eleme,~ts ~ all with decidaNe equalities--, have equality frvtgmen~,~ axiomafized by D E Q + (Lag), 3. Tl~e eql~ality consequences of the theories of inlcgraI do nains and iields wi~h deeidab!e eqtm!ities a:e axiomatized over D E Q by the sd~eme
where p i.s prime, k > ( L m > p ~ , and 0(m, iP) is the :|east p a w e r ~f g,> .m. T h e theory :is decidable~ M o ~ mundane exam)ples are given by considering the intuitionis~ic theories of fiehN of stmeified non-zero characteristi~.'s wiIh d e c | d a N e eqt;alitk~s. Even c'~assicaiiy one must add the axioms asserting that atf finite mo, l:~:!s have tile p,:o|'wr e~?:;'d~.ities: I ¢ p is the charg~c~eristic, the axioms arc
m arbLtrary as~;~ ~:, ~{s aL-<,ve. " I ~ e are the same axioms one needs to add to D E O to c:btair, the con~.q~;e~::,> ~:~:"~he ii~tuitio~istic tileories w'i~h decidable equa~it~es~
i~ this fi~]a| sectio,~, we wish to discuss various matt:crs a r i d cite an occasional co,pen" p r e b m m when it seems appropriate. We b e d n with a tech,licaI ma,{er~ T h e technique used in this paper is fairly general a n d hence ~s:~~e~y~q~-": ~,~ n~=t~a,wa}
considered the problen:~ at hand, A
:
c~as;~;;;:~:.:
classical .L-theory with o~Iy ~N~ite models~, tlaen the theory .r~-'~ e,~.e~d~,-~eT k,V a non-trivial L-aummorphis:m is ¢o,.~se~sdve ever T~ i ~ other words,, T :i'~e~:!: axiomatizes fl~e L-c'on,~eqttenc~s oa. *~ ,pad,t: Usiae the general ~-{e.~v:-<,f "- ° the autommphism by. {is ~-~p~rfox;ll,m:;~,~i~r., ~.,, .~ .,-,- woa.dd :,:~.,~f.:~L4tiiae axiO~:a?s c i t e d ;h-~ Sectioit 3 axioms :which, b y fhe way.. are still .~.~ec-:>ssarv~,: . . . . . . . . . Tile question arises: How necessary ,~,',:>°~" " ~ " .. . ~,~..,,.~. We have ix:en carehfl to demonsh°ate v; c,a~i~ythe tlee{:ssJly of the raew a:dOmatizations otlered ........that ~s; we have show~ by means of c:oumerexamp!es tl~a~ ~:~m:e new axioms were indeed needed whei~ we caffered them° Were a~.~ o~i '&era -
E~ma~ple. E Q ~L is not finitely axiomatized over EQ. gupF~ase it were. ~q ~_~e~ the equalRy fg~gment of G + ( D ) over D E Q wo,.fld ~ finitely axiomatizedo B u t it ~s clear the (Lag} is I~ot equivNent to any finite set of its if~stances. F\~r its non-finite axiomatizability prc
b y 13a:~wlse ~o n e ~s usuath'~ intc:res~:ed in c n t y c o u m a b i ~.......... ....... t),[.cs,-~~" a• ~l~e.,~ ~ a: v . u"~ " I n fact. he' finds t h a t oiie ofte~ ~t~¢ds o ~ : v ~o x~a~iz.¢, £~?c r~:c>..rsi~, ~\~,e~ \\-'~ !~ave used t m c o u n t a b i e m o d e l s for conveme,~c:e...... we ,.,........... ,~,~. u s e d com'..t:,b~e rood*., e[s a n d o n l y c o u u t a b N m a n y t:d?es It is no,: <~:a~ ~o~ e',.c~, taat we c o ' d d ae~ by with o n l y recursive s a m r a t i o ~ : T h e recm'sio.>t!:eo:'et~,c cotmte~>a~t l:o '~ .......... .... *{{..C, ~"" is a n i~crease in c o m p l e x i % F o r . if we are ~o realize t W e s me~tioe, in,~t decisions m a d e at earlier n o d e s , we are w o r k i n g rec~.?:stvdv b ~ h e &~cMcm~.; m a d e earuer. q~,,. ra:ises q,aest~o~s: D o we n e e d this cew'..pivt;ih~:? ~g ~'.oL.w ¢ cord<{ avi>id i{:c K ' C a~dd t h e s a ~ d a r d c.:>.~:structk~1 of sa~n~,<
notion
o f :oF-.si:ructive ~ r o v a b i ~ i i v .
obt~i~
but ~
"{'~e s a m e b o i d s o:[ \V~,h~siei~fs i'esu}.ts ~- -' '
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a~rcadv c - . , : u n n l t e d t o a Igtore ~It~cr; ~ m~.>iatheor'v. . . . . ¢ tb.e rea~s or snc\v ~ to be nou-a~omafir.abIe~ tn O A F we" s ~ e s , . ' e d tha! m o r e expe:dence in a x i ) n a t i z i n ~ the c o n s e q u e n c e s of various theories might give: us a c,cd~! perst'.egth~{~ oi~ the i[<:*" -~l~,:,~<,tt:.!l.\ v e }I :we m o r e exa! Ip!eG b u t .ci;tmlof dain~_ tch:we ,~ ~>~+.....p.er~oective: g h e j u m p s ~"I-v d-v,~- *,-~ " ~ 1.~i.,e 0 t v . ~ c ~...... l e h "-.2" ~, I l I I i [ ' O ~ T ~ liOElO};~eri{~1{v ...."-,.h,~,..... u ,~v.t~ geneiiy, t h o u g h n o t m~expeclcd fro ~ experience, w k h ,.~.~.t,,' "~,'" ,,,,~;'>_0;,~~troups, d o raise the ,:u~esliow \\'t~e~c is it all £oiu,::: ~,.~e n d ? D o {he toooi
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